Basic Functions - Linear Functions and Their Graphs | AI tutor The No.1 Homework Finishing Free App

'If the graph of y = f(x) on page 278 of book 146 is symmetrical with respect to the point (2,1), find the equation of the graph after shifting it parallelly -2 in the x-direction and -1 in the y-direction.'

'Change table: a table showing the increase or decrease of variables.'

'(3) (1) The proof is omitted, the condition for the equation to hold is x=y'

'Solve the following problems.'

'When the point (x, y) moves 79, find the region represented by the system of inequalities y ≤ 1/2x + 3, y ≤ -5x + 25, x ≥ 0, y ≥ 0. Find the maximum and minimum values of the following expressions:\n(1) x²+y²\n(2) x²+y²-2(x+6y)\n[Tokyo University of Science]'

'The condition for lines (2) and (3) to be parallel is'

'Plot the recurrence relation an+1=ran on a graph and explain its characteristics.'

'(1) For the line y=2x+3, find the coordinates of the point P(3,4) and its symmetrical point.'

"The equation of the tangent at the point (a, a²) on C₁ is y' = 2a, which leads to y - a² = 2a(x - a) thus, y = 2ax - a². The equation of the tangent at the point (b, 4b² + 12b) on C₂ is y' = 8x + 12, which leads to y - (4b² + 12b) = (8b + 12)(x - b) thus, y = (8b + 12)x - 4b². The line l coincides when (1) and (2) are equal, so 2a = 8b + 12, -a² = -4b² which gives a = 4b + 6 b² + 4b + 3 = 0 solving for b results in b = -1, -3 since the slope of line l is positive, 8b + 12 > 0 hence b = -1 a = 2. Therefore, the equation of line l is y = 4x - 4."

'Find the range of the function y=-2x+3 (-3 ≤ x ≤ 2).'

'Find the equation of the following lines:'

'Derive the equation of a straight line.'

'Plot the recurrence equation an+1=an+d on a graph and describe its characteristics.'

'Find the equation of the following lines.'

'The line y = mx (m > 0) bisects the area of shape T. Find the value of m.'

'When m = -4, it is 3,5; when m = 4, it is -5,-3 (2) When m = -2√5, it is √5, 3√5; when m = 2√5, it is -√5, -3√5'

'Find the equation of the line that is symmetric to the line y=2x+3 with respect to the line 3x+y-1=0.'

'Find the equation of a line passing through point (x_1, y_1) with slope m.'

'Find the equation of the line passing through the following two points: (1) (2,-3),(-1,1) (2) (3,4),(3,1) (3) (a, 0),(0, b) Line equation passing through different points (x1, y1),(x2, y2) y-y1=\x0crac{y2-y1}{x2-x1}(x-x1) when x coordinates of two points are different x1≠x2 when x coordinates of two points are equal x1=x2 the line equation is x=x1'

'Let a and b be real numbers, when a + b = 2 and a ≠ b, determine the order of 1, ab, and (a² + b²) / 2.'

'How is the shaded area on the right represented by a system of inequalities? Note that the boundary lines are not included in the region.'

'Find the definite integrals. (1) $\\int_{0}^{3}|x-2| dx$ (2) $\\int_{-2}^{3}\\left|x^{2}-2x\\right| dx$'

'Please explain the method of finding the general term from the given form of a recurrence relation.'

'Find the equations of the following lines:\n(1) Slope is -2, y-intercept is 3.\n(2) Passes through the point (4,2) with a slope of 3.\n(3) Passes through the point (-3,0) with a slope of -5.\n(4) Passes through the point (2,-1) with a slope of 1/2.\n(5) Passes through the point (-2,7) and is perpendicular to the x-axis.\n(6) Passes through the point (3,2) and is parallel to the x-axis.'

'Summary of arithmetic and geometric progressions'

'Increasing when x ≤ 2 and decreasing when 2 ≤ x. Always increasing. Decreasing when x ≤ -2/√3 or 2/√3 ≤ x, increasing when -2/√3 ≤ x ≤ 2/√3.'

'If the speed changes as shown in Figure 3, how much distance is covered in 5 hours?'

'As the time intervals of the graph in Figure 5 are shortened further, which part of the graph will the distance traveled within 5 hours fall on?'

'Question 2: In the context of life imagined as a line starting from birth and ending in death, where do you see yourself currently positioned?'

'What is a normal vector?'

'Express the equation of the line passing through the points (-3,2) and (2,-4) using the parameter t.'

'(1) \ y = x - 2\\sqrt{x} \'

'Express the equation of the straight line passing through the points (-3,2) and (2,-4) using the parameter t.'

'Find the range of the given functions. Also, determine the maximum and minimum values if they exist.'

'The range of the function y=ax+b (1≤x≤2) is 3≤y≤4.'

'In mathematics, for a in the interval [4, 6]:\n\nWhen a<4, the minimum value at x=a+1 is a^2-7a-9.\n\nWhen 4 ≤ a ≤ 5, the minimum value at x=5 is a-25.\n\nWhen a>5, the minimum value at x=a is a^2-9a.'

'Please explain how to utilize the sample list.'

'In general, the following holds. When the variable x is transformed by y=ax+b (a, b are constants), mean: ȳ=ax̄+b, variance: s_y^2=a^2s_x^2, standard deviation: s_y=|a|s_x'

'Investigate the functions F(a) and G(a) in the following domain:'

'What are the maximum and minimum of a function when one end of the domain moves?'

'Find the range of the following functions. Also, determine the maximum and minimum values.'

'When x + 3y = k, the minimum value of x^2 + y^2 is 4. Find the value of the constant k.'

"Please provide the page for 'linear function'."

'What is CHART (チャート)?'

'Find the range of the following functions. Also, determine the maximum and minimum values of the functions.'

'(1) Find the maximum value of xy when x + y = 4.'

'When the domain of a linear function y=ax+b is -3 ≤ x ≤ 1, the range is -1 ≤ y ≤ 3. Here, a>0.'

'Find the range of the function y=x-3 (1 ≤ x < 5).'

'When x=0, the minimum value is -1, and there is no maximum value.'

'Points of intersection of the graph and x-axis and real number solutions of the equation'

'Show the general form of a linear function and a quadratic function: where a, b, c are constants.'

'The minimum value is -1 when x=2, and there is no maximum value.'

'Plot the graph of the following functions and determine their range.'

'For x ≥ 0, y ≥ 0, and 3x + 2y = 1, find the maximum and minimum values of 3x^2 + 4y^2.'

'64 (1) The range is -5 ≤ y ≤ 4, the maximum value is 4 when x=-1, and the minimum value is -5 when x=2'

'Find the range of the following functions. Also, determine the maximum and minimum values of the functions.'

'Determine the values of the constants a and b such that the range of the linear function y=ax+b (-2 ≤ x ≤ 1) is -1 ≤ y ≤ 5. It is given that a<0.'

'Find the range of the following functions. Also, find the maximum and minimum values of the functions. (1) y=-3 x+1 (-1 ≤ x ≤ 2)(2) y=\\frac{1}{2} x+2 (-2<x ≤ 4)(3) y=-2 x^{2} (-1<x<1)'

'64 (2) The range is 1 < y ≤ 4, the maximum value is 4 when x=4, there is no minimum value'

'Find the minimum value of x^2 + y^2 when 2x + y = 3.'

'In the expression x+y=k, where we try to find the maximum and minimum values, we assume x+y=k and solve it as shown in example 106. The idea is as follows: For all values of (x, y) included in the domain D, calculating the value of x+y and finding the maximum and minimum values of x+y is impossible. Therefore...'

'Basic Example 73\nIntersection of 2 lines and solutions to a system of linear equations\nFind the conditions for the system of equations ax+3y-1=0,3x-2y+c=0 to have the following:\n(1) A unique solution\n(2) No solution\n(3) Infinite solutions'

'(2) If f(x) is a first-order function of x and \\int_{0}^{1} f(x) d x = 1, then \\int_{0}^{1}\\{f(x)\\}^{2} d x > 1.'

'In the xy-plane, let D be the region represented by the system of inequalities x ≥ 0, y ≥ 0, x+2y ≤ 30, 5x+2y ≤ 66. As the point (x, y) moves within region D, find the maximum value of kx+y, where k is a real number satisfying 1 ≤ k ≤ 3.'

'(1) $y = - \\frac{1}{2a} x - \\frac{1}{16 a^{2}}$\\n(2) Minimum value is $\\frac{1}{12}$ when $a = \\frac{1}{2}$'

'Choose one that fits the blank of E from the following (0) to (2).'

'Plot the regions represented by the following inequalities on the xy plane.'

'In mathematics, a problem to find the equation of the line PQ passing through the points of contact P, Q of two tangents.'

"(1) y' = -2x + 4 = -2(x - 2) When y' = 0, x = 2 the table of increase and decrease of y is as follows on the right. Therefore, increase for x <= 2 and decrease for x >= 2."

'Find the number of intersection points between the circle x^2 + y^2 = 1 and the following lines: (1) x + y = 1 (2) x - y = √2 (3) 2x - y + 5 = 0'

'What is the best way to proceed with working on example problems in step 2?'

'Since this line passes through the point (3, 4), we have 4 = -2(a - 2)・3 + a^2 - 3. Simplifying, we get a^2 - 6a + 5 = 0, which means (a - 1)(a - 5) = 0. Therefore, a = 1, 5. Hence, the equation of the required tangent line is y = 2x - 2 for a = 1 and y = -6x + 22 for a = 5.'

'Find the equation of the following lines: (1) Passes through the point (-3,5) with slope sqrt(3) (2) Passes through points (5,-3) and (-7,3) (3) Passes through points (5,1) and (3,2) (4) x-intercept at 4, y-intercept at -2 (5) Passes through points (-3,1) and (-3,-3) (6) Passes through points (1,-2) and (-5,-2)'

'In the xy plane, let D be the region defined by the system of inequalities x ≥ 0, y ≥ 0, x+2y ≤ 30, 5x+2y ≤ 66. Find the maximum value of kx+y as the point (x, y) moves within region D, where k is a real number satisfying 1 ≤ k ≤ 3.'

'Find the equation of the line parallel to the line ℓ: 2x+3y=4 and passing through the point (1,2). Also, find the equation of the line perpendicular to the line ℓ and passing through the point (2,3).'

'In a factory, there are two types of products, X and Y. To produce 1kg of X, 1kg of raw material A and 3kg of raw material B are required, while 2kg of raw material A and 1kg of raw material B are required for 1kg of Y. The maximum limits of available raw materials are 10kg for raw material A and 15kg for raw material B. If the profit per 1kg is 50,000 yen for X and 40,000 yen for Y, how many kilograms of X and Y should be produced to maximize profit?'

'Find the equation of the line passing through two different points $(x_1, y_1)$ and $(x_2, y_2)$.'

'At t = 1, maximum value is 2/3, at t = 1/2, minimum value is 1/4'

"(2) From $x-y=\\sqrt{2}$, we have $y=x-\\sqrt{2}$. Substituting this into $x^{2}+y^{2}=1$, we get \\[ x^{2}+(x-\\sqrt{2})^{2}=1 \\] Simplifying, we get $2x^{2}-2\\sqrt{2}x+1=0$. Let's denote the discriminant of this quadratic equation as $D$, then \\[ \\frac{D}{4}=(-\\sqrt{2})^{2}-2 \\cdot 1=0 \\] Since $D=0$, there is 1 point of intersection (tangent)."

"(1) Given y' = 6x -4 = 2(3x - 2), y'=0, when x=2/3, the table for the increase and decrease of y is as shown on the right. Therefore, y attains a minimum value of -1/3 at x=2/3."

'Solve the inequality 2|x+1|-|x-1|>x+2 using graphical methods.'

'Find the values of constants a and b such that the range of the function y = ax + b (-2 ≤ x < 1) is 1 < y ≤ 7.'

'In order for the point (2x-3, -3x+5) to be in the second quadrant, determine the range of values for x. Also, specify the quadrant where this point does not exist regardless of the value of x.'

'Find the values of the constants a and b such that the range of the function y = ax + b (2≤x≤5) is -1≤y≤5.'

"The complement of a set \ \\overline{A}=\\{x \\mid x \\in U \ and \ x \\notin A\\} \, De Morgan's laws state that \ \\overline{A \\cup B}=\\overline{A} \\cap \\overline{B} \\quad \\overline{A {cap B}=\\overline{A} \\cup \\overline{B} \"

'When the maximum value of the function y = -x + 1 (a ≤ x ≤ b) is 2, and the minimum value is -2, find the values of the constants a and b.'

'In the graph of the linear function y=ax+b, does it slope upwards to the right when a is positive or negative?'

'Find the range of the following functions. Also, if available, find the maximum and minimum values.'

'Find the values of constants a and b such that the range of the function y = ax + b (1 ≤ x ≤ 2) is 3 ≤ y ≤ 5.'

'Please describe the characteristics of the graph of the function y = b.'

'When the maximum value of the function y = -x + 1 (a ≤ x ≤ b) is 2 and the minimum value is -2, find the values of the constants a and b. It is known that a < b.'

'Determining the coefficients of a linear function based on the range conditions'

'Practice plotting the graphs of the following functions.'

'Graph of absolute value of a linear function (2)'

'Please describe the general form of a linear function and its shape on a graph.'

'When the proposition p → q is true, what conditions are met?'

'Let x be a variable with data consisting of n real numbers x_1, x_2, ..., x_n. Let the average of x_1, x_2, ..., x_n be denoted by x̄, and the standard deviation be denoted by s_x. When a new variable y and its data y_1, y_2, ..., y_n are defined by the expression y=4x-2, express the mean 𝑦̄ and the standard deviation s_y of y_1, y_2, ..., y_n using x̄ and s_x.'

'Find the values of constants a and b such that the range of the function y = ax + b (2 ≤ x ≤ 5) is -1 ≤ y ≤ 5.'

'A function that includes absolute value treats the case where the expression inside the absolute value becomes 0 as a point of division, and considers removing the absolute value symbol. In such cases of division points, it was learned in mathematics I example 67 that the graph of the function will curve. Here, we will examine how the graphs of such functions change when including absolute values and character constants.'

'Graph of an absolute value 1st-degree function (1)'

'When a=1, the graph becomes a line with a slope of 0 at x ≤ 1 and x ≥ 2.'

'Let x be a variable with mean x̄ and standard deviation s_x. When a new variable u is obtained by u=ax+b (where a and b are constants), and the data of u has a standard deviation s_u, then s_u=|a|s_x holds.'

'Let θ (0° < θ < 180°) be the angle to be found. (1) Since the slope of the line is -1, find θ. (2) Since the slope of the line is √3, find θ. (3) Since the slope of the line is -1/√3, find θ.'

'Determine the values of constants a and b to satisfy the following conditions: (1) For the linear function f(x)=ax+b, f(0)=-1 and f(2)=0. (2) The graph of the linear function y=ax+b passes through the points (-1,2) and (3,6). (3) When the domain of the function y=ax+b is -3≤x≤1, the range is 1≤y≤3. Here, a>0.'

'The score X is calculated based on the distance D.'

'Q10 Given, y= \\frac{9}{5} x+32. The average maximum temperature in Fahrenheit is y, and in Celsius is x. According to (1), y= \\frac{9}{5} x+32, so when \\ bar{x}=20, \\[ y= \\frac{9}{5} \\cdot 20+32= \\text{ A } 68\\left( ^{ °} \\text{ F} \\right) \\]'

'Find the range of the following functions. Also, determine the maximum and minimum values, if any. (1) y=x+2 (0 ≤ x ≤ 3) (2) y=4-2x (-1 ≤ x < 2)'

'The relationship between graphs and inequalities\nGenerally, when \\( f(x) > g(x) \\), it means that the graph of the function \\( y = f(x) \\) is above the graph of the function \\( y = g(x) \\). For example, in the inequality \ x+2 > |2x+1| \, you need to determine the range of values of \ x \ for which the graph of the function \ y = x+2 \ is above the graph of the function \ y = |2x+1| \. In the right diagram, the region highlighted in red represents this range, so the solution to the inequality is \ -1 < x < 1 \. This method can be effective in situations requiring complex analysis.'

'Find the acute angle θ formed by the following two lines.'

'In the coordinate plane, point P starts from the origin O and moves along the x-axis to point (6,0) at a speed of 1 unit per second, while point Q simultaneously starts from point (0,-6) and moves towards the origin O at a speed of 1 unit per second. At what time after departure will the distance between point P and point Q be minimized? Also, determine the minimum distance.'

'Find a linear function with the domain of \-2 \\leqq x \\leqq 2\ and the range of \-2 \\leqq y \\leqq 4\.'

'Try plotting the graph of the function y=x-[x](-2≤x≤3).'

'Regression Line'

'Determine the values of the constants a and b such that the range of the function y = ax + b (2 ≤ x ≤ 5) is -1 ≤ y ≤ 5. It is given that a < 0.'

'For the function f(x)=-2x+1, find the following values.'

'The following table summarizes the relationship between the price per piece of onigiri A and the sales volume at a company that manufactures and sells side dishes.'

'What is the main part of this book?'

'What kind of shape does the set of all points satisfying the equation |z+2|=2|z-1| represent?'

'Find the two points of intersection between the line (1) and the ellipse (2).'

''

'Since two tangents are perpendicular, \m_{1}m_{2} = -1\, therefore, \\\frac{v^{2}-a^{2}}{u^{2}+1} = -1\ leads to \u^{2} + v^{2} = a^{2} - 1\'

'(1) (A) f(x)=k x ( k is a constant )'

"What does it mean to reverse engineer from 'the self you want to become'?"

'Find the range of \ x \.'

'Find the angle θ between the following two planes. Let 0° ≤ θ ≤ 90°.'

'Find the solutions to the following:\n(1) Given y1 = 2p (x+x1), find y = -y1/(2p)(x-x1) + y1\n(2) Given y = 2/√3 x -1/√3, y = -√3/2 x + 2√3'

'Inverse Mapping\nFor a mapping f: A -> B, if f(A) = B and for every b in B there exists a unique a in A such that f(a) = b, then a mapping f^{-1}(b) = a can be defined from B to A. In other words, f(a) = b if and only if a = f^{-1}(b). f^{-1} is known as the inverse mapping of f. When the domain and codomain are both numbers (subsets of real numbers), the mapping is called a function. In other words, a mapping is a generalization of a function.'

'Example 9 | Composite functions and function determination'

'Find the equations of the tangent and normal lines.'

'(2) y = (6-x)/(x-2) = -1 + 4/(x-2) (1) y = (1/2)x + 1 The range of x values where the graph of function (1) is above the line (2), or where the graph of function (1) intersects with the line (2) is x ≤ -1 - √17, 2 < x ≤ -1 + √17'

'Question 92 General Rate of Change of Volume There is a container in the shape of an upside-down frustum as shown on the right. At a height of 4 cm, the horizontal cross-section is a square with side length 3 cm. When water is gently added to the container at a rate of 9 cm³ per second, at the moment when the depth of the water is 2 cm, what is the speed at which the water surface rises in centimeters per second? [Jichi Medical University]'

'Find the equation of the tangent line at the given point on the following curve.'

'Please prove the following equation: \n\\[\\alpha \ar{z}-\ar{\\alpha} z =(a+b i)(x-y i)-(a-b i)(x+y i) =2(b x-a y) i\\]\nIn particular, note that \\(\ar{\\alpha} z=(a x+b y)+(a y-b x) i\\) is not a real number, therefore, given that \ a y-b x \\neq 0 \, show that \ \\alpha \\overline{z}-\\overline{\\alpha} z \ is purely imaginary.'

'Consider the sphere S with diameter endpoints at two points A(0,3,0) and B(0,-3,0) in coordinate space. Find the maximum value of 3x + 4y + 5z as point P(x, y, z) moves on the sphere S. Also, determine the coordinates of P at that point.'

'When the two inequalities x+y-2 ≤ 0 and x^2+4x-y+2 ≤ 0 are satisfied, find the maximum and minimum values of (y-5)/(x-2), as well as the values of x and y at that time.'

'Find the area of the region represented by the system of inequalities 2y-x^2≥0, 5x-4y+7≥0, x+y-4≤0.'

'Plot the region represented by the following inequalities.'

'Find the conditions for the constants a, b such that f(x) <= 1 holds for all real numbers x, and plot the range of points (a, b) that satisfy these conditions.'

'Find the conditions for real numbers a, b such that the line y = ax + b has a point in common with the line segment connecting the 2 points A(-3,2) and B(2,-3), and represent it as a region in the ab plane.'

'Find the range of the function y = -2x + 1 (-1 ≤ x ≤ 2).'

"Given functions f(x), g(x), and their derivatives f'(x), g'(x), satisfying f(x)+g(x)=-2 x+5, f'(x)-g'(x)=-4 x+4, f(0)=5. Determine f(x) and g(x) in this case."

'Derive the equation of a line with slope m and y-intercept n, given by y = mx + n.'

'The lines (a-1)x-4y+2=0 and x+(a-5)y+3=0 intersect perpendicularly when a=〇, and are parallel when 80a=〇.'

'(1) Express S as a function of a when f(0)=0, f(2)=2.\n(2) Find the minimum value of S while satisfying f(0)=0, f(2)=2 as f varies.'

'Substitute (1) into $y = x + 2$ in (3) and simplify to get $x^2 + x - 6 = 0$ Therefore, $(x - 2)(x + 3) = 0$ hence $x = 2, -3$ From (3) we get $x = 2$ when $y = 4$, and $x = -3$ when $y = -1$ Therefore, the coordinates of the two points are $(2, 4), (-3, -1)$'

'Example (13) Equation of a straight line'

'Equation of a line, relationship between 2 lines'

'Assuming the function y=f(x) is continuous and a is a real constant. For all real numbers x, if the inequality |f(x)-f(a)|≤2/3|x-a| holds, prove using the intermediate value theorem that the curve y=f(x) must intersect the line y=x.'

'Find the limit of the sequence {an} defined by the following conditions.'

'What kind of curve is represented by the point P(x, y) in the following equations?'

'Find the coordinates and length of the chord formed by the intersection of the following line and curve: (1) y=3-2x, x^2+4y^2=4 (2) x+2y=3, x^2-y^2=-1'

"On the 'Summary' page, various topics learned from different places are summarized in a user-friendly manner on a single page."

'Please provide the equation of a circle with center α and radius r.'

'Find the coordinates of the points of intersection of the graphs of the line y = 8x-2 and the function y = √(16x-1).'

'Find all the linear functions g(x) that satisfy the conditions.'

'(1) Maximum value is 3/2 at x=-3, minimum value is -1/2 at x=1'

'1st independent and 1st dependent'

'Find the value of $x$ in Exercise 9 of Chapter 1 - Functions.'

'Practice (1) $\\ vec {a} = (- 1,2), \\ vec {b} = (2,4)$. When changing the value of the real number $t$, find the minimum value of the 2-norm of $\\ vec {c} = \\ vec {a} + t \\ vec {b}$, and the corresponding value of $t$.'

'For the functions f(x)=1-2x, g(x)=1/(1-x), h(x)=x(1-x), find the following composite functions.'

'Find the points of tangency when the equation of the tangent line is x=1 and the point of tangency is (1,0); and when the equation of the tangent line is y=\\frac{5}{2} x+\\frac{3}{2} and the point of tangency is \\left(-\\frac{5}{3},-\\frac{8}{3}\\right).'

'Graph of a representative function'

'(1) Show that f(0)=1.'

'Find the maximum and minimum values of 3x + 2y when real numbers x and y satisfy the two inequalities y ≤ 2x + 1, 9x² + 4y² ≤ 72.'

'How to draw a graph'

'How can you utilize the list of examples provided in each chapter?'

'Practice (1) When the line x-4=y-3=\\frac{z+2}{4} intersects the plane 2x+2y+z-2=0, let the smaller of the two angles formed be θ, find the value of cosθ.'

'Find the locus of points P that satisfy the following conditions:\n124 (1) The ratio of the distance from point F(4,2) to the line x=1 is 1:sqrt(2) for point P\n(2) The ratio of the distance from point F(0,-2) to the line y=3 is sqrt(6):1 for point P'

'Find the domain and range of Practice problem 1 in Chapter 1 Functions.'

''

'Example 9 | Composite Functions and Function Determination\nFor f(x)=2x+1 and g(x)=-2x+3, find the function h(x) that satisfies h(f(x))=g(x). It is assumed that h(x) is a polynomial function.\nExample 6'

'45 (1) \\ (4 \\cdot y=2 \\cdot 2(x+2) \\) that is, \\ (y=x+2 \\) (2) \\ (\\frac{1 \\cdot x}{3}+\\frac{2 \\cdot y}{6}=1\\) that is, \\ (x+y-3=0 \\) (3) \\ ( 2 \\cdot \\sqrt{2} \\cdot x-(-\\sqrt{3}) \\cdot y=1 \\) that is, \\ (2 \\sqrt{2} x+\\sqrt{3} y-1=0 \\)'

'Find the values of a, b, and c in Chapter 1 Functions - EXERCISES, problem 2.'

'Solving y=-2 x+3 for x gives x=-\\frac{1}{2} y+\\frac{3}{2}, by swapping x and y, the inverse function is y=-\\frac{1}{2} x+\\frac{3}{2}'

'A mapping is called a function when both the domain and the range are numbers (subsets of real numbers). In other words, a mapping is a generalization of a function.'

'68 (1) \\( y=\\frac{\\sqrt{2}}{4} x+\\frac{\\sqrt{2}}{2},(2, \\sqrt{2}) \\)'

'What does CHART & THINKING mainly indicate?'

'Determining the coefficients from the range'

'Please list the basic functionalities included in the educational materials lineup by Suken Shuppan.'

'Values of functions'

'When x=\\frac{5}{4}, y=-\\frac{1}{4}, the maximum value is \\frac{9}{8}; When x=0, y=1, the minimum value is -2'

'Let [a] denote the greatest integer that does not exceed the real number a. Draw the graph of the following functions: (1) y=-[x] (-3 ≤ x ≤ 2) (2) y=[2 x-1] (0 ≤ x ≤ 2)'

'When the variables x, y satisfy the condition x+2y=1, find the following: (1) The minimum value of x^2+y^2. (2) The maximum value of x^2+y^2 when x≥0, y≥0'

'Represent the largest integer that does not exceed a real number a as [a]. Draw the graph of the following functions: (1) y=2[x] (-2 ≤ x ≤ 1) (2) y=[2 x] (-2 ≤ x ≤ 1)'

'When the axis x=a is in the range of 2<x, that is, when a>2, it is minimized at x=2 from the graph on the right. The minimum value is \\[ f(2)=-8a+4\\]'

'What is a graph of the function y = f(x)?'

'When the line x=a is in the range x<0, i.e., when a<0, it reaches the minimum at x=0 on the right graph. The minimum value is f(0)=-4a'

'\\[\egin{array}{c}f(x)=2 x \\\\ \\text{from the graph of } f(x) \\\\ 0 \\leqq f(x)<\\frac{1}{2} \\\\\n\\text{Therefore, } \\\\\nf(f(x))=2 f(x)=2 \\cdot 2 x\\end{array}\\]\nTherefore\n• is the graph of (1).\nThe line—below the equation \ y=\\frac{1}{2} \ doubles the parts below it, and the parts above (or on the line) double and then subtract 1.'

'Find the maximum and minimum values of P when x is between 0 and 3 and y is between 0 and 3.'

'Find the points of intersection between the graph of a function with an absolute value and a straight line.'

'Exercise 35→Book p.92\nLet f(x)=1/2 x-|x-|x-1||. The equation f(x)=a has three different solutions when the graph of y=f(x) and the line y=a have three different intersection points. Let g(x)=|x-|x-1||.\n[1] When x>=1, g(x)=|x-(x-1)|=1, therefore f(x)=1/2 x-g(x)=1/2 x-1.\n[2] When x<1, g(x)=|x+(x-1)|=|2x-1|\n(i) When x<1/2, g(x)=-2x+1, so f(x)=1/2 x-g(x)=1/2 x-(-2x+1)=5/2 x-1.\n(ii) When 1/2 <= x < 1, g(x)=2x-1, so f(x)=1/2 x-g(x)=1/2 x-(2x-1)=-3/2 x+1.\nTherefore, the graph of y=f(x) is as shown in the right figure. The range of values of a for which the graph and the line y=a have three different intersection points is -1/2<a<1/4.'

'When x = 1, y = 0, the maximum value is 2; when x = -1, y = 0, the minimum value is -2'

'When a=1, (0,1), when a=5, (-2,-1)'

'Find the range and maximum and minimum values'

'Please graph the function y=|x+1|+|x-3|.'

Determine the range of the following functions. Also, find the maximum and minimum values of the functions. (1) y=-3x+1 \quad (-1 \leqq x \leqq 2) (2) y=\frac{1}{2}x+2 \quad (-2<x \leqq 4) (3) y=-2x^{2} \quad (-1<x<1)

Given $x \geq 0, y \geq 0, x + y = 2$, find the maximum and minimum values of $x^2 + y^2$.

Development 78 | Graph of a linear function involving absolute value

Maximum and Minimum with Conditions (1) Given $x \geqq 1, y \geqq -1, 2 x + y = 5$, find the maximum and minimum values of $x y$.

For the linear function \( f(x)=a x+b \), given \( f(1)=2 \) and \( f(3)=8 \), find the values of the constants $a$ and $b$.

Standard 65 | Determining the coefficients of a linear function from conditions such as the range

When buying items costing 100 yen each, the total cost depends on the quantity purchased. Similarly, when a car travels at 60 km/h, the travel time determines the total distance. Let's learn about such relationships where 'one quantity determines another quantity'. Function definition: For example, when you buy x items each costing 100 yen, the total cost y in yen can be expressed as y=100x. In such a case, when there are two variables x and y, if determining one value of x uniquely determines the value of y, we say y is a function of x. A function is like a factory that takes the input quantity x, processes it by 'multiplying x by 100', and produces the output result y.

Find the range of the function y=x−3 \quad(1 \leqq x<5).

Find the minimum value of $x^{2} + y^{2}$ when $2x + y = 3$.

(2) For the linear function $y=ax+b$ with the domain $-3 \leqq x \leqq 1$, the range is $-1 \leqq y \leqq 3$. Assume $a>0$.

When the domain of the function \( y=f(x) \) is $1 \leqq x \leqq 5$, it is shown as follows. \[ y=f(x) \quad(1 \leqq x \leqq 5) \] At this point, what is the range of the function?

Determine the values of the constants $a$ and $b$ so that the range of the linear function $y = ax + b$ is $-1 \leqq y \leqq 5$. Assume $a < 0$.

There is a rectangle with a perimeter of 20 cm. If the length of the rectangle is x cm and the area is y cm², then y is a function of x. Answer the following questions. (1) Express y in terms of x and state the domain of this function. (2) When this function is f(x), find f(3), f(1/2), and f(a+1).

Find the coordinates of the midpoint and the length of the segment cut by the ellipse (x-2)^2 + 4(y-4)^2 = 4 from the line x+2y=11.

When the coordinate plane is regarded as the complex plane, points on the plane correspond to complex numbers. Additionally, various calculations between complex numbers can be illustrated on the plane. Therefore, among the problems related to plane figures, some can be solved more easily and clearly by using properties or calculations of complex numbers. Here, we summarize the basics of using complex numbers in problems involving plane figures. Internal and external division points of a line segment Let α = x1 + y1i, β = x2 + y2i, z = x + yi, and if point P(z) internally divides line segment AB in the ratio m:n, then (x - x1) : (x2 - x) = (y - y1) : (y2 - y) = m:n Thus, x = (n x1 + m x2) / (m + n), y = (n y1 + m y2) / (m + n) Hence, z = x + yi = (n x1 + m x2) / (m + n) + (n y1 + m y2) / (m + n)i = (n (x1 + y1i) + m (x2 + y2i)) / (m + n) = (n α + m β) / (m + n) Similarly, the external division can be considered as well. Therefore, the following holds: if point C(γ) internally divides line segment AB in the ratio m:n and point D(δ) externally divides in the ratio m:n, then Internal division point γ = (n α + m β) / (m + n) External division point δ = (- n α + m β) / (m - n) Specifically, the complex number representing the midpoint of line segment AB is (α + β) / 2.

Let the line $y=2x+k$ intersect the hyperbola $x^{2}-y^{2}=1$ at two distinct points $P$ and $Q$. (1) Find the range of possible values for the constant $k$. (2) When $k$ varies within the range found in (1), determine the locus of the midpoint $M$ of the segment $PQ$.