Question

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) =

Explanation:

Step1: Recall property of inverse - function graph

The graph of $y = f^{-1}(x)$ is the reflection of the graph of $y = f(x)$ about the line $y=x$.

Step2: Find two - points on $y = f(x)$

From the graph of $y = f(x)$, we can see that the line $y = f(x)$ passes through the points $(0,1)$ and $(1,3)$. So the points on $y = f^{-1}(x)$ will be $(1,0)$ and $(3,1)$ after reflection about $y = x$. Sketch the line passing through these points for $y = f^{-1}(x)$.

Step3: Use the formula for the derivative of the inverse function

The formula for the derivative of the inverse function is $(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}$. First, we need to find $x$ such that $f^{-1}(-3)=x$, which means $f(x)=-3$. From the graph of $y = f(x)$, when $y=-3$, $x = - 2$. So $f^{-1}(-3)=-2$.

Step4: Find the slope of $y = f(x)$

The function $y = f(x)$ is a linear function. The slope of the line $y = f(x)$ using two - points $(x_1,y_1)=(0,1)$ and $(x_2,y_2)=(1,3)$ is $m = f'(x)=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 1}{1 - 0}=2$.

Step5: Calculate $(f^{-1})'(-3)$

Since $(f^{-1})'(-3)=\frac{1}{f'(f^{-1}(-3))}$ and $f^{-1}(-3)=-2$ and $f'(x) = 2$ (for all $x$ as $f(x)$ is linear), then $(f^{-1})'(-3)=\frac{1}{f'(-2)}=\frac{1}{2}$.

Answer:

$\frac{1}{2}$