Question

the following function is one-to-one. find the inverse of the function and graph the function and its inverse on the same set of axes.
$f(x)=2x - 3$
$f^{-1}(x)=\square$
(type a simplified fraction.)
use the graphing tool to graph the function and its inverse.
click to enlarge graph

Explanation:

Step1: Replace f(x) with y

We start by writing the function as \( y = 2x - 3 \). This is a standard step when finding the inverse of a function, as it helps us manipulate the equation to solve for \( x \) in terms of \( y \).

Step2: Swap x and y

Next, we interchange the roles of \( x \) and \( y \) to get \( x = 2y - 3 \). This is based on the definition of an inverse function, where if \( f(a)=b \), then \( f^{-1}(b)=a \), so swapping the input and output variables is a key step.

Step3: Solve for y

Now we solve the equation \( x = 2y - 3 \) for \( y \). First, we add 3 to both sides of the equation: \( x + 3 = 2y \). Then, we divide both sides by 2 to isolate \( y \): \( y=\frac{x + 3}{2} \). We can also write this as \( y=\frac{1}{2}x+\frac{3}{2} \).

Step4: Replace y with \( f^{-1}(x) \)

Finally, we replace \( y \) with the inverse function notation \( f^{-1}(x) \). So, \( f^{-1}(x)=\frac{x + 3}{2} \).

Answer:

\( f^{-1}(x)=\frac{x + 3}{2} \)