Ortiz Pearson
04/18/2024 · Middle School
Algebraic For the following exercises, find the inverse of the function on the given domain. 5. \( f(x)=(x-4)^{2},[4, \infty) \) 6. \( f(x)=(x+2)^{2},[-2, \infty) \) 7. \( f(x)=(x+1)^{2}-3,[-1, \infty) \) 8. \( f(x)=2-\sqrt{3+x} \) 9. \( f(x)=3 x^{2}+5, \quad(-\infty, 0] \) 10. \( f(x)=12-x^{2},[0, \infty) \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the inverse of the given functions on the specified domains, we will follow these steps:
1. Replace \( f(x) \) with \( y \).
2. Solve for \( x \) in terms of \( y \).
3. Swap \( x \) and \( y \) to express the inverse function.
Let's go through each function one by one.
### 5. \( f(x) = (x - 4)^2, [4, \infty) \)
1. Replace \( f(x) \) with \( y \):
\[
y = (x - 4)^2
\]
2. Solve for \( x \):
\[
\sqrt{y} = x - 4 \quad \text{(since \( x \geq 4 \), we take the positive root)}
\]
\[
x = \sqrt{y} + 4
\]
3. Swap \( x \) and \( y \):
\[
f^{-1}(x) = \sqrt{x} + 4, \quad [0, \infty)
\]
### 6. \( f(x) = (x + 2)^2, [-2, \infty) \)
1. Replace \( f(x) \) with \( y \):
\[
y = (x + 2)^2
\]
2. Solve for \( x \):
\[
\sqrt{y} = x + 2 \quad \text{(since \( x \geq -2 \), we take the positive root)}
\]
\[
x = \sqrt{y} - 2
\]
3. Swap \( x \) and \( y \):
\[
f^{-1}(x) = \sqrt{x} - 2, \quad [0, \infty)
\]
### 7. \( f(x) = (x + 1)^2 - 3, [-1, \infty) \)
1. Replace \( f(x) \) with \( y \):
\[
y = (x + 1)^2 - 3
\]
2. Solve for \( x \):
\[
y + 3 = (x + 1)^2
\]
\[
\sqrt{y + 3} = x + 1 \quad \text{(since \( x \geq -1 \), we take the positive root)}
\]
\[
x = \sqrt{y + 3} - 1
\]
3. Swap \( x \) and \( y \):
\[
f^{-1}(x) = \sqrt{x + 3} - 1, \quad [3, \infty)
\]
### 8. \( f(x) = 2 - \sqrt{3 + x} \)
1. Replace \( f(x) \) with \( y \):
\[
y = 2 - \sqrt{3 + x}
\]
2. Solve for \( x \):
\[
\sqrt{3 + x} = 2 - y
\]
\[
3 + x = (2 - y)^2
\]
\[
x = (2 - y)^2 - 3
\]
3. Swap \( x \) and \( y \):
\[
f^{-1}(x) = (2 - x)^2 - 3, \quad (-\infty, 2]
\]
### 9. \( f(x) = 3x^2 + 5, (-\infty, 0] \)
1. Replace \( f(x) \) with \( y \):
\[
y = 3x^2 + 5
\]
2. Solve for \( x \):
\[
y - 5 = 3x^2
\]
\[
x^2 = \frac{y - 5}{3}
\]
\[
x = -\sqrt{\frac{y - 5}{3}} \quad \text{(since \( x \leq 0 \))}
\]
3. Swap \( x \) and \( y \):
\[
f^{-1}(x) = -\sqrt{\frac{x - 5}{3}}, \quad [5, \infty)
\]
### 10. \( f(x) = 12 - x^2, [0, \infty) \)
1. Replace \( f(x) \) with \( y
Quick Answer
### 5. \( f^{-1}(x) = \sqrt{x} + 4, [0, \infty) \)
### 6. \( f^{-1}(x) = \sqrt{x} - 2, [0, \infty) \)
### 7. \( f^{-1}(x) = \sqrt{x + 3} - 1, [3, \infty) \)
### 8. \( f^{-1}(x) = (2 - x)^2 - 3, (-\infty, 2] \)
### 9. \( f^{-1}(x) = -\sqrt{\frac{x - 5}{3}}, [5, \infty) \)
### 10. \( f^{-1}(x) = \sqrt{12 - y}, [0, \infty) \)
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit