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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![## Composite Functions Evaluation Using Graphs
This educational segment will guide you on evaluating composite functions using a graph. Specifically, we will address the evaluations from (a) to (f) based on the given graph. Let's begin with part (a).
### Part (a): Evaluating (f ∘ g)(1)
The given graph contains two functions:
- \( g(x) \) represented by a red line.
- \( f(x) \) represented by a blue curve.
To find \( (f \circ g)(1) \), follow these steps:
1. **Find \( g(1) \)**:
- Locate \( x = 1 \) on the x-axis of the graph.
- Determine the corresponding y-value on the red graph of \( g(x) \).
- According to the graph, \( g(1) = 3 \).
2. **Evaluate \( f(g(1)) \)**:
- Now that we know \( g(1) = 3 \), we need to find \( f(3) \).
- Locate \( x = 3 \) on the x-axis of the graph.
- Determine the corresponding y-value on the blue graph of \( f(x) \).
- According to the graph, \( f(3) = 1 \).
Thus, \( (f \circ g)(1) = f(g(1)) = f(3) = 1 \).
**Answer box:**
\[ (f \circ g)(1) = \boxed{1} \]
### Graph Explanation:
- **Axes and Scaling**: The horizontal axis (x-axis) is marked from -6 to 6, and the vertical axis (y-axis) is also marked from -6 to 6.
- **Red Line (g(x))**: This is a straight line with a positive slope, intersecting the y-axis at \( y = -1 \).
- **Blue Curve (f(x))**: This is a parabolic curve starting at \( x = -4 \) and ending at \( x = 4 \), peaking at \( y = 4 \) when \( x = 0 \).
In subsequent parts, other similar evaluations will be addressed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ae5415e-4394-4e91-afd5-48b91660ab22%2F4be6ca48-e1d7-45ec-b5e7-8624e4035042%2F7tov9pg_processed.png&w=3840&q=75)
Transcribed Image Text:## Composite Functions Evaluation Using Graphs
This educational segment will guide you on evaluating composite functions using a graph. Specifically, we will address the evaluations from (a) to (f) based on the given graph. Let's begin with part (a).
### Part (a): Evaluating (f ∘ g)(1)
The given graph contains two functions:
- \( g(x) \) represented by a red line.
- \( f(x) \) represented by a blue curve.
To find \( (f \circ g)(1) \), follow these steps:
1. **Find \( g(1) \)**:
- Locate \( x = 1 \) on the x-axis of the graph.
- Determine the corresponding y-value on the red graph of \( g(x) \).
- According to the graph, \( g(1) = 3 \).
2. **Evaluate \( f(g(1)) \)**:
- Now that we know \( g(1) = 3 \), we need to find \( f(3) \).
- Locate \( x = 3 \) on the x-axis of the graph.
- Determine the corresponding y-value on the blue graph of \( f(x) \).
- According to the graph, \( f(3) = 1 \).
Thus, \( (f \circ g)(1) = f(g(1)) = f(3) = 1 \).
**Answer box:**
\[ (f \circ g)(1) = \boxed{1} \]
### Graph Explanation:
- **Axes and Scaling**: The horizontal axis (x-axis) is marked from -6 to 6, and the vertical axis (y-axis) is also marked from -6 to 6.
- **Red Line (g(x))**: This is a straight line with a positive slope, intersecting the y-axis at \( y = -1 \).
- **Blue Curve (f(x))**: This is a parabolic curve starting at \( x = -4 \) and ending at \( x = 4 \), peaking at \( y = 4 \) when \( x = 0 \).
In subsequent parts, other similar evaluations will be addressed.
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