the following is the graph of a function $f(x)$. a. on the same axes, sketch a graph of the inverse function…

the following is the graph of a function $f(x)$. a. on the same axes, sketch a graph of the inverse function $f^{-1}(x)$. b. use the graph of the inverse to evaluate $(f^{-1})(3)$. $(f^{-1})(3)=$ question help: video

the following is the graph of a function $f(x)$. a. on the same axes, sketch a graph of the inverse function $f^{-1}(x)$. b. use the graph of the inverse to evaluate $(f^{-1})(3)$. $(f^{-1})(3)=$ question help: video

Answer

Explanation:

Step1: Find the value of (x) such that (f(x)=3)

From the graph, when (y = f(x)=3), (x = 1.5). So (f(1.5)=3), which implies (f^{-1}(3)=1.5)

Step2: Use the formula ((f^{-1})^{\prime}(a)=\frac{1}{f^{\prime}(f^{-1}(a))})

First, find the slope of (y = f(x)). The function (y=f(x)) is a line. Using two - point formula (m=\frac{y_2 - y_1}{x_2 - x_1}). Let's take two points ((- 0.5,-1)) and ((1.5,3)). Then (m=\frac{3+1}{1.5 + 0.5}=\frac{4}{2}=2), so (f^{\prime}(x)=2) (since (f(x)) is linear, its derivative is constant) Since (a = 3) and (f^{-1}(3)=1.5), and (f^{\prime}(x)=2) for all (x) (because (f(x)) is a linear function (y = 2x-1), (f^{\prime}(x)) is the slope of the line). Then ((f^{-1})^{\prime}(3)=\frac{1}{f^{\prime}(f^{-1}(3))}) Substitute (f^{-1}(3)=1.5) into the formula: ((f^{-1})^{\prime}(3)=\frac{1}{f^{\prime}(1.5)}) Since (f^{\prime}(x) = 2) (the slope of the linear function (f(x))), ((f^{-1})^{\prime}(3)=\frac{1}{2})

Answer:

(\frac{1}{2})