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## Functions and Analysis

### Basic Functions - Rational Functions and Inequalities

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'Practice graphing the region represented by the following inequality.'

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'Find the minimum value of x+\\frac{9}{x} when x>0.'

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'By altering the positions of the sound sources, we measured the difference in the time the sound was heard at A and B, and found that there is a maximum at that time.'

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'Plot the graphs of the functions (1) $y=\\frac{1}{2x}$, (2) $y=\\frac{3}{x}-1$, and (3) $y=\\frac{-2}{x-1}$.'

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'Determining Coefficients of a Function from Maximum and Minimum Values (2)'

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'Basic 1: Graph of a rational function, asymptotes, and range'

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'Let C1 be the curve symmetric with respect to the line y=x for the curve y=2/(x+1), and let C2 be the curve symmetric with respect to the line y=-1 for the curve y=2/(x+1). Find all coordinates of the intersection points between the asymptote of curve C2 and curve C1.'

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'By studying the existence of real solutions of the equation $f(x)=g(x)$,\nand investigating the value changes of the function $F(x)=f(x)-g(x)$, the intermediate value theorem is used.\n(1) If $F(x)$ is continuous on the closed interval $[a, b]$, and $F(a)F(b)<0$ [$F(a)$ and $F(b)$ have opposite signs], then there is at least one real solution for the equation $F(x)=0$ in the open interval $(a, b)$.\n(2) In (1), in particular, if $F(x)$ is monotonically increasing [$F^{\\prime}(x)>0$] or monotonically decreasing [$F^{\\prime}(x)<0$], then the real solution is unique.'

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'Please sketch the graph of the function y=(1-log x)/x^2. It is important to note that lim(x→∞) of log x/x^2 is equal to 0.'

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'Basic 3: Intersection of the graphs of a fraction function and a line, fraction inequalities'

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'How is the graph of the function y =\\ frac {-6 x +21} {2 x-5} moved in parallel to the graph of the function y =\\ frac {8 x +2} {2 x-1}?'

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'The graph of the function y = (ax + b) / (x + 2) (b ≠ 2a) passes through the point (1, 1), and the inverse function of this function is the same as the original function. Find the values of constants a, b.'

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'Let z be a non-zero complex number. When the inequality 2 ≤ z + 16/z ≤ 10 holds, plot the region where point z exists in the complex plane.'

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'Understand the range of a function and conquer example 64!'

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'Explain the calculation of expressions involving square roots.'

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'When x=\\frac{6}{5}, y=\\frac{3}{5}, the minimum value is \\frac{9}{5}.'

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'Explain the converse, contrapositive, and inverse of a proposition.'

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'Find the minimum value of x+\\frac{16}{x} when x>0.'

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'Prove that for positive numbers a, b, x, and y with a+b=1, √(ax+by) ≥ a√(x)+b√(y). Also, determine when the equality holds.'

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'Explain the relationship between sets and necessary and sufficient conditions.'

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'Please explain sets and related terms (subset, equal, intersection, union, complement).'

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'(4) Choose one of the following 0-7, assuming that the graphs of y=|3x-6| and y=2x+1 are represented on the same coordinate plane, the most appropriate one for their intersection points.'

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'The function expression can be transformed into the form y=k/(x-2)+1, and since the graph passes through the point (1,2), from 2=-k+1 we get k=-1'

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'Find the value of the following definite integral.\ \\int_{0}^{1} \\frac{1}{x^{2}+x+1} d x \'

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'If functions f(x) and g(x) are continuous at the value α in their domain, prove that the following functions are also continuous at x=α: 1. k f(x) + l g(x) (where k, l are constants) 2. f(x) g(x) 3. f(x)/g(x) (where g(α) ≠ 0)'

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'164 (1) u = \\frac{V}{2 \\pi} \\cdot \\frac{1-2 h+\\sqrt{1-4 h}}{h^{2}}'

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'Graphs, domains, and ranges of 3 irrational functions'

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'Consider g(1/2) = α, g(1/3) = β, and prove that α + β = π/4.'

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'Find the values of constants $a, b, c$ when the function $f(x)=\\frac{a x+b}{x+c}$ satisfies the following conditions (A), (B).\n(A) The curve $y=f(x)$ intersects with the line $y=x+1$ at two points, and the absolute values of the $x$ coordinates of these two intersection points are equal.\n(B) The curve $y=f(x)$ and the intersections with the $x$-axis and $y$-axis are both on the line $y=\\frac{3}{2} x+\\frac{11}{2}$.\n[Keio University]'

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'Find the value of the following definite integral. \\[(1)\\]\y = \\frac{1}{\\sqrt{x}} to x = \\frac{1}{y^2}\\\\frac{1}{2} \\le y \\le 1\\text{, }x > 0\\]\\The required definite integral is \\[S = \\int_{\\frac{1}{2}}^{1}\\frac{d y}{y^2} = \\left[-\\frac{1}{y}\\right]_{\\frac{1}{2}}^{1}\'

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'Plot the graph of the function y = \\frac{9x - 10}{6x - 4} and find the asymptotes.'

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'y = \\frac{9x-10}{6x-4} = \\frac{9x-10}{2(3x-2)} = \\frac{3(3x-2)-4}{2(3x-2)} = \\frac{3}{2} - \\frac{2}{3x-2}. Therefore, the graph we seek is obtained by translating the graph of y = \\frac{-2/3}{x} 2/3 units along the x-axis and 3/2 units along the y-axis to the right. The graph is shown in the diagram to the right. The asymptotic lines are the two lines x=2/3, y=3/2.'

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'Example 36: Approximate formulas and approximate values\n(1) Create the first-order and second-order approximation formulas of f(x)=\\frac{1}{1+x} when |x| is sufficiently small.\n(2) Using the first-order approximation formula of \\cos(a+h), find the approximate value of \\cos 61 degrees. Take \\sqrt{3}=1.732, \\pi=3.142, and calculate to the third decimal place.'

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'Given the fractional function f(x) = \\frac{a x-b}{x-2}, where b \\neq 2 a. For all x satisfying 0 \\leqq x \\leqq 1, it is required that 0 \\leqq f(x) \\leqq 1 and f(f(x))=x. Find the values of constants a, b.'

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'Graph, asymptote, and range of the rational function.'

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'Approximate the following functions.\n(1) Calculate 1/(1+x) ≈ 1-x, 1/(1+x) ≈ 1-x+x^2 in order\n(2) 0.485'

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"Let y = \\ frac{2 x + 9}{x + 2} = \\ frac{5}{x + 2} + 2 \\ n y = - \\ frac{x}{5} + k \\ n, If we define the number of points at which the graph of function (1) and the line (2) intersect to be equal to the number of real solutions given by the equation. \\ n \\ n \\ frac{2 x + 9}{x + 2} = - \\ frac{x}{5} + k, then 5(2 x + 9) = -x(x + 2) + 5 k(x + 2) \\ Rearranging, we get \\ n x ^ {2} + (12-5 k) x + 5(9-2 k) = 0 Let's denote the discriminant as D: \\ n D = (12-5 k) ^ {2} - 4 \\ cdot 1 \\ cdot 5(9-2 k) = 25 k ^ {2} -80 k-36 = (5 k + 2) (5 k -18) \\ n Setting D = 0, we get k = - \\ frac{2}{5}, \\ frac{18}{5} In this case, the graph of function (1) and line (2) intersect. Hence, the number of real solutions we are looking for is: 2 solutions for k < - \\ frac{2}{5}, \\ frac{18}{5} <k; 1 solution for k = - \\ frac{2}{5}, \\ frac{18}{5}; no solutions for - \\ frac{2}{5} <k < \\ frac{18}{5}"

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'Find the range of constant a when a tangent line can be drawn from the point (a, 0) to the graph of the function y=(x+3)/(√(x+1)).'

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'(1) How is the graph of the function y = \\frac{3 x+17}{x+4} related to the graph of the function y = \\frac{x+8}{x+3} after a parallel shift? (2) Determine the values of the constants a, b, c when the graph of the function y = \\frac{a x+b}{x+c} has the asymptotes x = 3 and y = 1, and passes through the point (2, 2).'

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'For a real number x, [x] represents the integer n satisfying n ≤ x < n+1, determine the values of constants a and b such that the function f(x)=( [x] + a)( b x - [x] ) is continuous at x=1 and x=2.'

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'Find the equation of the tangent line drawn from the given point P to the curve and the coordinates of the point of tangency. (i) y=1/x+1, P(1,-2)'

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'Find the equation of the asymptotes of y = \\frac{x^{3}}{x^{2}-4}.'

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'Plot the graphs of the following functions and determine their asymptotes.'

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'Plot the graphs of the following functions and find their asymptotes. (a) y=(3x+5)/(x+1) (b) y=(-2x+5)/(x-3) (c) y=(x-2)/(2x+1) (2) Find the range of values for functions (a) and (b) when the domain is 2 ≤ x ≤ 4.'

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'Investigate the continuity of the following functions and specify their domains.'

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'Next, we investigate the number of real solutions of the equation based on the intersections of the graph of y=f(x) and the line y=k.'

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'(1) y=\\frac{4x-3}{x-2} y=5x-6 From (1) and (2) we get \\frac{4x-3}{x-2}=5x-6 Multiplying both sides by x-2 we get 4x-3=(5x-6)(x-2) Simplifying we get x^2-4x+3=0 Therefore (x-1)(x-3)=0 Hence x=1,3 Substituting into (2) we have y=-1 when x=1 and y=9 when x=3 Therefore, the coordinates of the intersection points are (1, -1), (3, 9)'

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'Find the coordinates of the points where the graph of the function f(x) = 1/6x³ + 1/2x + 1/3 intersects with the graph of its inverse function f^{-1}(x).'

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'Find the equations of the asymptotes for (1) y=\\frac{2 x^{2}+3}{x-1} (2) y=x-\\sqrt{x^{2}-9}'

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'When the function is y=(2x+c)/(ax+b), passes through the point (-2, 9/5), and has the asymptotes x=-1/3, y=2/3, find the values of the constants a, b, c.'

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'Let C: y = 1/x (x > 0) be the curve. Consider the tangent line to the point P(t, 1/t) on the curve C as lt. Moreover, let α, β be constants satisfying 0 < α < β, and let D be the region enclosed by the two tangent lines lα, lβ, and the curve C. (1) Find the area of D. (2) For α < t < β, find the value of t that minimizes the area S(t) above the tangent line lt in D.'

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'Investigate if the following functions are continuous and differentiable at the points specified within [ ].'

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'In the range of \ 0 \\leqq x \\leqq 2 \\pi \, the function \\( f(x) \\) is defined as \\( f(x)=\\frac{2 a(\\sin x+\\cos x)}{2+2 \\sin x \\cos x-a(\\sin x+\\cos x)} \\). Here, \ a \ is a constant satisfying \ 0<a<2 \.'

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'(1) Find the coordinates of the intersection points of the graph of the function \ y=\\frac{2}{x+3} \ and the line \ y=x+4 \. \n(2) Solve the inequality \ \\frac{2}{x+3}<x+4 \.'

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'For the interval \ [a, b] \ where \ a < b \, if \\( f(x) \\neq g(x) \\) and \\( f(x) \\geqq g(x) \\) for all \ x \, then \\( \\int_{a}^{b} f(x) dx > \\int_{a}^{b} g(x) dx \\)'

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'Draw the graph of the fraction function and asymptotic line, in addition to solving the basic example 73 (1) function y = 3x / (x-2). Also, find the asymptotic line. (2) In (1), when the domain is 4 ≤ x ≤ 8, find the range.'

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'Find the area S enclosed by the graphs y=1/x, y=ax, y=bx on the xy plane. Here, x>0, a>b>0.'

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'Investigate the increase and decrease of the function y=(x²-x+2)/(x+1), the concavity of the graph, the asymptotes, and draw a rough sketch of the graph.'

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'When real numbers a, b, c, d satisfy ad-bc≠0, answer the following questions regarding the function f(x)=\\frac{ax+b}{cx+d}:'

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'Investigate the increase and decrease, concavity and convexity, asymptotic lines of the function y=(x+1)^{3}/x^{2}, and sketch the outline of the graph.'

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'From the conditions of the asymptotes, the desired function can be expressed as $y = \\frac{k}{x+3} + q (k \\neq 0)$. Since the graph passes through the points $(-2,3)$ and $(1,6)$, we have the equations $3=k+q, 6=\\frac{k}{4}+q$. Solving these equations gives $k=-4, q=7$. Substituting these values, we get $y=\\frac{-4}{x+3}+7$, which simplifies to $y=\\frac{7x+17}{x+3}$. The asymptotes of the graph of $y=\\frac{k}{x-p}+q$ are two lines $x=p, y=q$.'

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'Asymptotic lines not parallel to both axes (y = ax + b)'

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'Prove that the function f(x)=\\frac{x}{1+2^{\\frac{1}{x}}} when A 46^{\\ominus} x \\neq 0 and f(x)=0 when x=0 is continuous at x=0 but not differentiable.'

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'Translate the given text into multiple languages.'

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'Find the values of the constants a, b, and c when the graph of the function y = \\frac{a x + b}{2 x + c} passes through the point (1, 2) and has the asymptotic lines x = 2 and y = 1.'

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'The domain of the function (1) is x ≠ -p/2, and the range is y ≠ 1/2. Therefore, the domain of the inverse function of (1) is x ≠ 1/2. In order for function (1) and its inverse function to match, -p/2 must equal 1/2, so p = -1. In this case, the inverse function of (1) is y = (x + 4) / (2x - 1), which matches function (1). Therefore, the value of the required constant p is p = -1.'

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'Plot the graphs of the following functions and find their domain and range.'

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'Examine the function for increasing and decreasing intervals, concavity of the graph, asymptotes, and sketch the general shape of the graph.'

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'Find the value of the constant a such that the inverse function of the function y=(a x-a+3)/(x+2) is equal to the original function (a is not equal to 1)'

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'Determine the values of the constants p, q so that the function f(x) = (px + q)/(x^2 + 3x) attains a local maximum of -9 at x = -1/3.'

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'What curve does the given parametric representation represent?'

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'Determine whether the following functions f(x) are continuous or discontinuous. Where [x] denotes the greatest integer that does not exceed the real number x.'

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'Explain the relationship between the inverse function g(y) and the original function f(x).'

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'Find the equation of the tangent line when a point (x₁, y₁) lies on the curve of the ellipse $\\frac{x²}{a²} + \\frac{y²}{b²} = 1$.'

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'Graph of a rational function, asymptotes, and range'

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'The domain of function (1) is x ≠ -2, the range is y ≠ a, thus the domain of the inverse function of (1) is x ≠ a, in order for function (1) and its inverse function to match\na=-2\nAt this time, the inverse function of (1) is y = \\frac{-2x+5}{x+2}, which matches function (1). Therefore, the value of the constant a that is sought is a=-2'

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'Plot the graph of the function y=(-2x+7)/(x-3) for x in the interval [1, 4] and determine its range.'

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'Plot the graph of the function y=(9x-10)/(6x-4) and find the asymptotes.'

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'Prove that the function $f(x)=\\frac{x}{1+2^{\\frac{1}{x}}}$ when $x \\neq 0$ and $f(x)=0$ when $x=0$ is continuous at $x=0$ but not differentiable.'

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'Determine the values of the constants a, b such that the function f(x)=\\frac{a x+b}{x^{2}+1} attains a maximum value of \\frac{1}{2} at x=\\sqrt{3}.'

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Basic 64 | Range of functions, maximum and minimum values of functions (Basic)

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Please explain the properties of the following hyperbola: x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0)