escOGRAPHS AND FUNCTIONSQuotient of two functions: AdvancedSuppose that the functions fand g are defined as follows.53f(x) = ² 8(x)= x+8FindgThen, give its domain using an interval or union of intervals.Simplify your answers.()-DDomain ofExplanation1ggCheck2#34%5[0,0](0,0)-80XAOVO (0,0)Ø6State Un...S(0,0)8&7Home+Ⓒ2023 McGraw8

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Answered: Suppose that the functions fand g are…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 52E

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Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. Suppose g(x)=f(x) where f(x)0 for all x. Explain why the local minima and maxima of f and g occur at the same values of x. Let gx be the distance between the point 3,0 and the point (x,x2) on the graph of the parabola y=x2. Express g as a function of x. Find the minimum value of the function g that you found in part b. Use the principle described in part a to simplify your work.
A triangle is formed by the coordinate axes and a line through the point (2,1), as shown in the figure. The value of y is given by y=1+2x2 Write the area A of the triangle as a function of x. Determine the domain of the function in the context of the problem. Sketch the graph of the area function. Estimate the minimum area of the triangle from the graph.
F1x2 = - x4 + 8x2 + 9(a) Determine whether F is even, odd, or neither.(b) There is a local maximum value of 25 at x = 2. Determinea second local maximum value.(c) Suppose the area under the graph of F betweenx = 0 and x = 3 that is bounded from below by thex-axis is 50.4 square units. Using the result from part (a),determine the area under the graph of F betweenx = - 3 and x = 0 that is bounded from below by thex-axis.
f(x) = x3 - 2x2 - 15x + 36 = (3 - x)(12 - x - x2) Find the (a) intervals on which f is decreasing and increasing as well as the critical values of f(x), if any; (b) coordinates of local maximum and minimum values (if any); (c) intervals on which f is concave up and concave down as well as the points of inflection, if any; Sketch the graph of and show all details on the graph.
Let f(x)=x^2−4/x^2−1 f′(x)=6x/(x^2−1)^2 f′′(x)=−6(3x^2+1)/(x^2−1)^3 (please enter response for that info above) a) Find all x- and y-intercepts and state the domain of the function. If there aren't any, state that there is none. b) Find the vertical and horizontal asymptotes of the function f(x). (Show all work using limits). c) Find the critical points of f(x). Find the open interval(s) where f(x) is increasing and the open interval(s) where f(x) is decreasing. If there aren't any of these, clearly state so. (show all work). d) State the x and y coordinates of any local maxima and local minima. If there aren't any, state that there is none. e) Find the open interval(s) where f(x) is concave up, and where f(x) is concave down. Find any inflection points. If there aren't any of these, clearly state so. (show all work). f) Sketch the curve, and label on your curve all the information found in part a - e.
A function is a ratio of quadratic functions and has a verticalasymptote x =4 and just one -intercept, x =1 . It isknown f that has a removable discontinuity at x =- 1 and lim x->-1f(x) = 2. Evaluate a) f (0) b) limx->∞f(x)
3.The absolute maximum and minimum values of f(x)=−(x^2+x)^2/3 over the interval [-3, 4]. absolute maximum is= and it occurs at x = absolute minimum is= and it occurs at x = Notes: If there is more than one x value, enter as a comma separated list .
A graph that is large enough to show a function's global behavior may fail to reveal important local features. The graph of f(x) = x^8/8-x^6/2-x^5+5x^3 is a case in point.a. Graph ƒ over the interval -2.5 <= x <= 2.5, where does the graph appear to have local extreme values or points of inflection?b. Now factor f '(x) and show that f has a local maximum at x = Cuberoot[5] and local minima at x = [PlusMinus] Sqrt[3].c. Zoom in on the graph to find a viewing window that shows the presence of the extreme values at x = Cuberoot[5] and x = [PlusMinus] Sqrt[3].
Two functions f(x) and g(x), are continuous and differentiable for all real numbers. Some values of the functions and their deravativee are given in the table above. Based on the table, what is the value of h'(2) if h(x) = f(g(x))?
Functions of the form f(x) = 5 · bkx for k = ±1 will be examined to study the effect of the parameter b on the graph. (a) Graph the function f(x) =5 · 2x. Use the graph to determine the y-values at x-values of −2, −1, 0, 1, and 2. For every increase of 1 in the x-value, the y-value can be found from the previous y-value by ---Select--- the addition of a constant multiplication by a constant . The constant is equal to . (b) Graph the function f(x) = 5 · 0.5x. Use the graph to determine the y-values at x-values of −2, −1, 0, 1, and 2. For every increase of 1 in the x-value, the y-value can be found from the previous y-value by ---Select--- the addition of a constant multiplication by a constant . The constant is equal to . (c) Graph the function f(x) = 5 · 2−x. Use the graph to determine the y-values at x-values of −2, −1, 0, 1, and 2. For every increase of 1 in the x-value, the y-value can be found from the previous y-value by ---Select--- the addition of a…
Use a computer algebra system to analyze the graph of the function. f(x) = 1/ x2 − 2x − 8 Label any extrema that exist. (If an answer does not exist, enter DNE.) relative maximum (x, y) = relative minimum (x, y) = Label any asymptotes that exist. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.) vertical asymptote(s) horizontal asymptote(s)
The function f(x) = 2x^3 - 30x^2 + 126x + 9 has one local minimum and one local maximum. Use a graph of the function to estimate these local externa. The function has a local minimum at x = _________ with output value __________ and a local maximum at x = ____________ with output value ______________

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Suppose that the functions fand g are defined as follows. 5 3 f(x)= 8(x)=x²8 g Find Then, give its domain using an interval or union of intervals. f Simplify your answers.