Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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part c
![A function \( f \) is given.
\[ f(x) = \sqrt{x} + 4 \]
(a) Sketch the graph of \( f \).
**Graph Details:**
The correct graph for the function \( f(x) = \sqrt{x} + 4 \) is in the bottom left corner, marked with a check mark.
- **Axes:**
- The horizontal axis is labeled as \( x \).
- The vertical axis is labeled as \( y \).
- **Graph Characteristics:**
- The graph starts at the point \((0, 4)\), since \( f(x) = \sqrt{x} + 4 \) is defined for \( x \geq 0 \).
- The graph extends to the right in a gradual curve upwards, representing the increase in function value as \( x \) increases.
The other graphs do not represent the given function \( f(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25e588b9-b272-406a-8435-8ef7027d16be%2F9855e77a-ddb9-4b55-9ee9-7ba34419b129%2Fys7gedi_processed.png&w=3840&q=75)
Transcribed Image Text:A function \( f \) is given.
\[ f(x) = \sqrt{x} + 4 \]
(a) Sketch the graph of \( f \).
**Graph Details:**
The correct graph for the function \( f(x) = \sqrt{x} + 4 \) is in the bottom left corner, marked with a check mark.
- **Axes:**
- The horizontal axis is labeled as \( x \).
- The vertical axis is labeled as \( y \).
- **Graph Characteristics:**
- The graph starts at the point \((0, 4)\), since \( f(x) = \sqrt{x} + 4 \) is defined for \( x \geq 0 \).
- The graph extends to the right in a gradual curve upwards, representing the increase in function value as \( x \) increases.
The other graphs do not represent the given function \( f(x) \).
![The image contains a math problem involving inverse functions. The task is to sketch the graph of the inverse function \( f^{-1} \) using the graph of \( f \).
### Graphs Description:
There are four graphs displayed:
1. **Top Left Graph:**
- This graph has an exponential shape with the curve starting from the upper left quadrant and moving downward to the right, passing near the y-axis.
2. **Top Right Graph:**
- The graph shows an exponential-like curve that starts near the x-axis in the center-right section and moves upward sharply.
3. **Bottom Left Graph:**
- This graph depicts a parabolic curve opening upwards, symmetric with respect to the y-axis, resembling the graph of \( y = x^2 \).
4. **Bottom Right Graph:**
- A curved line starts at the top right and moves diagonally downward to the left, resembling the graph of \( f(x) = -\sqrt{x} \).
### Instructions:
- **(b)**: Use the graph of \( f \) to sketch the graph of \( f^{-1} \).
- **(c)**: Find \( f^{-1} \).
The notation given is:
\[
f^{-1}(x) = \text{\_\_\_\_\_} \quad , \quad x \geq 0
\]
- A checkbox beside the bottom right graph is marked correctly, indicating that it represents the correct inverse function sketch.
These graphs and instructions suggest a problem about inverse functions, asking students to identify transformations and find the correct inverse function graphically and algebraically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25e588b9-b272-406a-8435-8ef7027d16be%2F9855e77a-ddb9-4b55-9ee9-7ba34419b129%2Fcj70rb6k_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains a math problem involving inverse functions. The task is to sketch the graph of the inverse function \( f^{-1} \) using the graph of \( f \).
### Graphs Description:
There are four graphs displayed:
1. **Top Left Graph:**
- This graph has an exponential shape with the curve starting from the upper left quadrant and moving downward to the right, passing near the y-axis.
2. **Top Right Graph:**
- The graph shows an exponential-like curve that starts near the x-axis in the center-right section and moves upward sharply.
3. **Bottom Left Graph:**
- This graph depicts a parabolic curve opening upwards, symmetric with respect to the y-axis, resembling the graph of \( y = x^2 \).
4. **Bottom Right Graph:**
- A curved line starts at the top right and moves diagonally downward to the left, resembling the graph of \( f(x) = -\sqrt{x} \).
### Instructions:
- **(b)**: Use the graph of \( f \) to sketch the graph of \( f^{-1} \).
- **(c)**: Find \( f^{-1} \).
The notation given is:
\[
f^{-1}(x) = \text{\_\_\_\_\_} \quad , \quad x \geq 0
\]
- A checkbox beside the bottom right graph is marked correctly, indicating that it represents the correct inverse function sketch.
These graphs and instructions suggest a problem about inverse functions, asking students to identify transformations and find the correct inverse function graphically and algebraically.
Expert Solution
Step 1
We plug f(x)=y then switch x and y.
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