Question

using equations to compare linear functions
consider the function and its inverse:
$f(x) = \frac{1}{3}x - 2$
$f^{-1}(x) = a(x + 2)$
the slope, $a$, of the inverse function is \boxed{}, and the $x$-intercept of the inverse function is at $x = \boxed{}$.

Explanation:

Step 1: Find the inverse of \( f(x) = \frac{1}{3}x - 2 \)

Let \( y = \frac{1}{3}x - 2 \). Swap \( x \) and \( y \): \( x = \frac{1}{3}y - 2 \).
Solve for \( y \):
Add 2 to both sides: \( x + 2 = \frac{1}{3}y \).
Multiply both sides by 3: \( y = 3(x + 2) \). So, \( f^{-1}(x) = 3(x + 2) = 3x + 6 \).

Step 2: Determine the slope \( a \) of \( f^{-1}(x) \)

The function \( f^{-1}(x) = 3x + 6 \) is in slope - intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( a = 3 \).

Step 3: Find the \( x \) - intercept of \( f^{-1}(x) \)

The \( x \) - intercept occurs when \( y = 0 \). Set \( f^{-1}(x)=0 \):
\( 0 = 3x + 6 \)
Subtract 6 from both sides: \( - 6 = 3x \)
Divide both sides by 3: \( x=-2 \)

Answer:

The slope \( a \) of the inverse function is \( \boldsymbol{3} \), and the \( x \) - intercept of the inverse function is at \( x=\boldsymbol{-2} \).