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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![### Finding the Inverse Function
To find the inverse function of \( f(x) = -\sqrt{x + 9} \):
1. **Replace \( f(x) \) with \( y \):**
\[
y = -\sqrt{x + 9}
\]
2. **Swap \( x \) and \( y \):**
\[
x = -\sqrt{y + 9}
\]
3. **Solve for \( y \):**
\[
-x = \sqrt{y + 9}
\]
\[
x^2 = y + 9
\]
\[
y = x^2 - 9
\]
Therefore, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = x^2 - 9
\]
### Specifying the Domain
To find the domain of \( f^{-1}(x) \):
Consider the domain of the original function \( f(x) = -\sqrt{x + 9} \). The expression inside the square root, \( x + 9 \), must be non-negative, so:
\[
x + 9 \geq 0
\]
\[
x \geq -9
\]
Since the original function \( f(x) \) outputs non-positive values (the negative square root), the range of \( f(x) \) (which becomes the domain of \( f^{-1}(x) \)) is:
\[
(-\infty, 0]
\]
Thus, the domain of \( f^{-1}(x) \) using interval notation is:
\[
(-\infty, 0]
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90d1e96e-989a-4849-aeb9-585c1e797a4d%2Ff19a91ee-5093-4985-a635-1d379975b45d%2F7ys2c3x_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Inverse Function
To find the inverse function of \( f(x) = -\sqrt{x + 9} \):
1. **Replace \( f(x) \) with \( y \):**
\[
y = -\sqrt{x + 9}
\]
2. **Swap \( x \) and \( y \):**
\[
x = -\sqrt{y + 9}
\]
3. **Solve for \( y \):**
\[
-x = \sqrt{y + 9}
\]
\[
x^2 = y + 9
\]
\[
y = x^2 - 9
\]
Therefore, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = x^2 - 9
\]
### Specifying the Domain
To find the domain of \( f^{-1}(x) \):
Consider the domain of the original function \( f(x) = -\sqrt{x + 9} \). The expression inside the square root, \( x + 9 \), must be non-negative, so:
\[
x + 9 \geq 0
\]
\[
x \geq -9
\]
Since the original function \( f(x) \) outputs non-positive values (the negative square root), the range of \( f(x) \) (which becomes the domain of \( f^{-1}(x) \)) is:
\[
(-\infty, 0]
\]
Thus, the domain of \( f^{-1}(x) \) using interval notation is:
\[
(-\infty, 0]
\]
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