For the real-valued functions f(x)=√5x+30 and g(x)=x-3, find the composition fg and specify its domain using interval notation. 0° 0/0 (f-g)(x) = [] 0101 (0,0) (0,0) OVO (0,0] [0,0) Domain of f g 0 8 -8 ? 00 Q X S
For the real-valued functions f(x)=√5x+30 and g(x)=x-3, find the composition fg and specify its domain using interval notation. 0° 0/0 (f-g)(x) = [] 0101 (0,0) (0,0) OVO (0,0] [0,0) Domain of f g 0 8 -8 ? 00 Q X S
Algebra and Trigonometry (6th Edition)
6th Edition
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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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![**Composition of Functions and Domain**
For the real-valued functions \( f(x) = \sqrt{5x + 30} \) and \( g(x) = x - 3 \), find the composition \( f \circ g \) and specify its domain using interval notation.
### Solution:
- To find the composition \( (f \circ g)(x) \), we substitute \( g(x) \) into \( f(x) \):
\[ (f \circ g)(x) = f(g(x)) = f(x - 3) \]
\[ f(x - 3) = \sqrt{5(x - 3) + 30} \]
\[ f(x - 3) = \sqrt{5x - 15 + 30} \]
\[ f(x - 3) = \sqrt{5x + 15} \]
Therefore,
\[ (f \circ g)(x) = \sqrt{5x + 15} \]
### Domain of \( f \circ g \):
- The expression inside the square root \( 5x + 15 \) must be non-negative for the function to be defined.
\[ 5x + 15 \geq 0 \]
\[ 5x \geq -15 \]
\[ x \geq -3 \]
- Hence, the domain of \( f \circ g \) in interval notation is:
\[ [-3, \infty) \]
### Graphs or Diagrams:
- This problem includes a matrix/diagram to select the correct expression and interval:
- The expression \( \sqrt{5x + 15} \)
- Various interval choices: \( \emptyset \), \( [0, 0] \), \( [0, \infty) \), \( (-∞, ∞) \), \( (-∞, 0] \), \( \infty - ∞ \)
From the work shown above, the correct expression is \( \sqrt{5x + 15} \) and the correct domain is \( [-3, \infty) \).
Please ensure that your students understand these steps to fully grasp the process of finding function compositions and determining their domains.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d7590ab-1e8b-4f01-9940-34be16d61a30%2F8e5ba5a8-05ee-49c3-9d64-6cf52a50b89d%2Fu4c8zxu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Composition of Functions and Domain**
For the real-valued functions \( f(x) = \sqrt{5x + 30} \) and \( g(x) = x - 3 \), find the composition \( f \circ g \) and specify its domain using interval notation.
### Solution:
- To find the composition \( (f \circ g)(x) \), we substitute \( g(x) \) into \( f(x) \):
\[ (f \circ g)(x) = f(g(x)) = f(x - 3) \]
\[ f(x - 3) = \sqrt{5(x - 3) + 30} \]
\[ f(x - 3) = \sqrt{5x - 15 + 30} \]
\[ f(x - 3) = \sqrt{5x + 15} \]
Therefore,
\[ (f \circ g)(x) = \sqrt{5x + 15} \]
### Domain of \( f \circ g \):
- The expression inside the square root \( 5x + 15 \) must be non-negative for the function to be defined.
\[ 5x + 15 \geq 0 \]
\[ 5x \geq -15 \]
\[ x \geq -3 \]
- Hence, the domain of \( f \circ g \) in interval notation is:
\[ [-3, \infty) \]
### Graphs or Diagrams:
- This problem includes a matrix/diagram to select the correct expression and interval:
- The expression \( \sqrt{5x + 15} \)
- Various interval choices: \( \emptyset \), \( [0, 0] \), \( [0, \infty) \), \( (-∞, ∞) \), \( (-∞, 0] \), \( \infty - ∞ \)
From the work shown above, the correct expression is \( \sqrt{5x + 15} \) and the correct domain is \( [-3, \infty) \).
Please ensure that your students understand these steps to fully grasp the process of finding function compositions and determining their domains.
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