First, let's analyze the given functions:
The function is:
[tex]\[ f(x) = \frac{1}{3}x - 2 \][/tex]
The inverse function is given in the form:
[tex]\[ f^{-1}(x) = a(x + 2) \][/tex]
To find the slope [tex]\( a \)[/tex] of the inverse function, let's first determine the actual form of the inverse function of [tex]\( f(x) \)[/tex].
### Finding the Inverse Function:
1. Start with the equation for [tex]\( f(x) \)[/tex], and let [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = \frac{1}{3}x - 2 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y + 2 = \frac{1}{3}x \][/tex]
[tex]\[ 3(y + 2) = x \][/tex]
[tex]\[ x = 3y + 6 \][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ f^{-1}(x) = 3(x + 2) \][/tex]
### Comparing with the Given Form:
The inverse function is given as [tex]\( f^{-1}(x) = a(x + 2) \)[/tex]. By comparison:
[tex]\[ 3(x + 2) \quad \text{and} \quad a(x + 2) \][/tex]
We can see that:
[tex]\[ a = 3 \][/tex]
### Finding the x-intercept:
To find the x-intercept of the inverse function [tex]\( f^{-1}(x) \)[/tex], set the function equal to 0 and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = 3(x + 2) \][/tex]
[tex]\[ 0 = 3x + 6 \][/tex]
[tex]\[ 3x = -6 \][/tex]
[tex]\[ x = -2 \][/tex]
Therefore, the slope [tex]\( a \)[/tex] of the inverse function is [tex]\( 3 \)[/tex] and the x-intercept of the inverse function is at [tex]\( x = -2 \)[/tex].
