4) Find the inverse of the function f(x)=√√x+5. Also write out the check. Then graph both and label each one. Also draw in the y = x line as a dashed line. Work: Check:
4) Find the inverse of the function f(x)=√√x+5. Also write out the check. Then graph both and label each one. Also draw in the y = x line as a dashed line. Work: Check:
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Find the inverse of the fuction. WRITE OUT A CHECK. Graph and both and label each one. Draw y = x line as dashed.
Problem attached- please follow all directions.
![### Finding and Verifying the Inverse of a Function
**Problem Statement**: Find the inverse of the function \( f(x) = \sqrt{x + 5} \). Also, write out the check. Then graph both the function and its inverse on the provided coordinate plane and label each one. Additionally, draw the line \( y = x \) as a dashed line.
**Work**:
1. **Finding the Inverse**:
- Start with the given function: \( y = \sqrt{x + 5} \).
- Swap \( x \) and \( y \) to find the inverse: \( x = \sqrt{y + 5} \).
- Solve for \( y \):
\[
x^2 = y + 5
\]
\[
y = x^2 - 5
\]
- Therefore, the inverse function is \( f^{-1}(x) = x^2 - 5 \).
2. **Check**:
- To verify that \( f(x) \) and \( f^{-1}(x) \) are indeed inverses, we need to check:
\[
f(f^{-1}(x)) = \sqrt{(x^2 - 5) + 5} = \sqrt{x^2} = x
\]
\[
f^{-1}(f(x)) = (\sqrt{x + 5})^2 - 5 = x + 5 - 5 = x
\]
- Both compositions return \( x \), confirming that \( f(x) \) and \( f^{-1}(x) \) are valid inverse functions.
**Graph**:
- The graph consists of three elements:
- The original function \( f(x) = \sqrt{x + 5} \).
- The inverse function \( f^{-1}(x) = x^2 - 5 \).
- The line \( y = x \) to check the symmetry, drawn as a dashed line.
**Coordinate Plane Details**:
- The provided coordinate plane is a standard Cartesian grid.
- To graph \( f(x) = \sqrt{x + 5} \), plot points for \( x \) and their corresponding \( y \) values (e.g., for \( x = -5, y = 0 \); for \( x = -4, y \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29666072-4841-4557-93a0-541aeee2aafd%2F297412cc-959f-4408-8e97-6e162bce5171%2Fxmrwirwq_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding and Verifying the Inverse of a Function
**Problem Statement**: Find the inverse of the function \( f(x) = \sqrt{x + 5} \). Also, write out the check. Then graph both the function and its inverse on the provided coordinate plane and label each one. Additionally, draw the line \( y = x \) as a dashed line.
**Work**:
1. **Finding the Inverse**:
- Start with the given function: \( y = \sqrt{x + 5} \).
- Swap \( x \) and \( y \) to find the inverse: \( x = \sqrt{y + 5} \).
- Solve for \( y \):
\[
x^2 = y + 5
\]
\[
y = x^2 - 5
\]
- Therefore, the inverse function is \( f^{-1}(x) = x^2 - 5 \).
2. **Check**:
- To verify that \( f(x) \) and \( f^{-1}(x) \) are indeed inverses, we need to check:
\[
f(f^{-1}(x)) = \sqrt{(x^2 - 5) + 5} = \sqrt{x^2} = x
\]
\[
f^{-1}(f(x)) = (\sqrt{x + 5})^2 - 5 = x + 5 - 5 = x
\]
- Both compositions return \( x \), confirming that \( f(x) \) and \( f^{-1}(x) \) are valid inverse functions.
**Graph**:
- The graph consists of three elements:
- The original function \( f(x) = \sqrt{x + 5} \).
- The inverse function \( f^{-1}(x) = x^2 - 5 \).
- The line \( y = x \) to check the symmetry, drawn as a dashed line.
**Coordinate Plane Details**:
- The provided coordinate plane is a standard Cartesian grid.
- To graph \( f(x) = \sqrt{x + 5} \), plot points for \( x \) and their corresponding \( y \) values (e.g., for \( x = -5, y = 0 \); for \( x = -4, y \
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