Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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].
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].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.