Question

consider this function.
$f(x) = 2x - 2$
which graph represents the inverse of function $f$?
(images of four graphs labeled w, x, y, z, and options a. w, b. z, c. y, d. x)

Explanation:

Step1: Find the inverse function

To find the inverse of \( f(x) = 2x - 2 \), we first replace \( f(x) \) with \( y \):
\( y = 2x - 2 \)
Then, we swap \( x \) and \( y \):
\( x = 2y - 2 \)
Next, we solve for \( y \):
Add 2 to both sides: \( x + 2 = 2y \)
Divide both sides by 2: \( y=\frac{x + 2}{2}=\frac{1}{2}x + 1 \)
So the inverse function is \( f^{-1}(x)=\frac{1}{2}x + 1 \)

Step2: Analyze the inverse function's graph

The inverse function \( f^{-1}(x)=\frac{1}{2}x + 1 \) is a linear function with a slope of \( \frac{1}{2} \) (positive, so the line is increasing) and a y - intercept of 1.

Now let's analyze the given graphs:

• Graph W: It has a positive slope and a y - intercept of 1, which matches the equation of the inverse function \( y=\frac{1}{2}x + 1 \).
• Graph X: It has a negative slope, so it can't be the graph of the inverse function (since our inverse function has a positive slope).
• Graph Y: Its slope and intercept do not match the inverse function.
• Graph Z: It has a negative slope, so it can't be the graph of the inverse function.

Answer:

A. W