Using Equations to Compare Linear Functions

Consider the function and its inverse:
[tex]\[ f(x)=\frac{1}{3} x-2 \quad f^{-1}(x)=a(x+2) \][/tex]

The slope, [tex]\(a\)[/tex], of the inverse function is [tex]\(\square\)[/tex] and the [tex]\(x\)[/tex]-intercept of the inverse function is at [tex]\(x=\)[/tex] [tex]\(\square\)[/tex].



Answer :

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].