Chapter10: Exponential And Logarithmic Functions
Section10.1: Finding Composite And Inverse Functions
Problem 64E: Explain how to find the inverse of a function from its equation. Use an example to demonstrate the...
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![### Understanding and Finding the Inverse Function
#### Given Function:
Let
\[ f(x) = \frac{x + 3}{x + 6} \]
To determine \( f^{-1}(-2) \), we first need to find the inverse of the function \( f(x) \).
#### Steps to Find the Inverse:
1. **Express \( y \) in terms of \( x \):**
\[ y = \frac{x + 3}{x + 6} \]
2. **Swap \( x \) and \( y \):**
\[ x = \frac{y + 3}{y + 6} \]
3. **Solve for \( y \) in terms of \( x \):**
\[ x(y + 6) = y + 3 \]
\[ xy + 6x = y + 3 \]
\[ xy - y = 3 - 6x \]
\[ y(x - 1) = 3 - 6x \]
\[ y = \frac{3 - 6x}{x - 1} \]
So, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \frac{3 - 6x}{x - 1} \]
#### Evaluate \( f^{-1}(-2) \):
Substitute \( -2 \) into the inverse function:
\[ f^{-1}(-2) = \frac{3 - 6(-2)}{-2 - 1} \]
\[ f^{-1}(-2) = \frac{3 + 12}{-3} \]
\[ f^{-1}(-2) = \frac{15}{-3} \]
\[ f^{-1}(-2) = -5 \]
Therefore,
\[ f^{-1}(-2) = -5 \]
By working through these steps, we determined that \( f^{-1}(-2) \) equals \(-5\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d8b0e94-47ef-477e-9622-abc8b98247d8%2F7c79a0f9-e4d7-4a69-86ae-4d73e2aed215%2Fuo73vyt_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding and Finding the Inverse Function
#### Given Function:
Let
\[ f(x) = \frac{x + 3}{x + 6} \]
To determine \( f^{-1}(-2) \), we first need to find the inverse of the function \( f(x) \).
#### Steps to Find the Inverse:
1. **Express \( y \) in terms of \( x \):**
\[ y = \frac{x + 3}{x + 6} \]
2. **Swap \( x \) and \( y \):**
\[ x = \frac{y + 3}{y + 6} \]
3. **Solve for \( y \) in terms of \( x \):**
\[ x(y + 6) = y + 3 \]
\[ xy + 6x = y + 3 \]
\[ xy - y = 3 - 6x \]
\[ y(x - 1) = 3 - 6x \]
\[ y = \frac{3 - 6x}{x - 1} \]
So, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \frac{3 - 6x}{x - 1} \]
#### Evaluate \( f^{-1}(-2) \):
Substitute \( -2 \) into the inverse function:
\[ f^{-1}(-2) = \frac{3 - 6(-2)}{-2 - 1} \]
\[ f^{-1}(-2) = \frac{3 + 12}{-3} \]
\[ f^{-1}(-2) = \frac{15}{-3} \]
\[ f^{-1}(-2) = -5 \]
Therefore,
\[ f^{-1}(-2) = -5 \]
By working through these steps, we determined that \( f^{-1}(-2) \) equals \(-5\).
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