Question
For f left parenthesis x right parenthesis equals StartRoot x EndRoot and g left parenthesis x right parenthesis equals x plus 4 comma find the following functions.
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesis; b. left parenthesis g circle f right parenthesis left parenthesis x right parenthesis; c. left parenthesis f circle g right parenthesis left parenthesis 5 right parenthesis; d. left parenthesis g circle f right parenthesis left parenthesis 5 right parenthesis
Question content area bottom
Part 1
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
enter your response here
(Simplify your answer.)
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesis; b. left parenthesis g circle f right parenthesis left parenthesis x right parenthesis; c. left parenthesis f circle g right parenthesis left parenthesis 5 right parenthesis; d. left parenthesis g circle f right parenthesis left parenthesis 5 right parenthesis
Question content area bottom
Part 1
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
enter your response here
(Simplify your answer.)
Answers
GPT-4o mini
To find the composite functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \) based on the given functions \( f(x) = \sqrt{x} \) and \( g(x) = x + 4 \), we'll perform the following steps:
### Part 1: Finding \( (f \circ g)(x) \)
1. **Definition of Composite Function**: The composite function \( (f \circ g)(x) \) is defined as \( f(g(x)) \).
2. **Calculate \( g(x) \)**:
\[
g(x) = x + 4
\]
3. **Substitute \( g(x) \) into \( f(x) \)**:
\[
f(g(x)) = f(x + 4) = \sqrt{x + 4}
\]
Thus, the expression for \( (f \circ g)(x) \) is:
\[
(f \circ g)(x) = \sqrt{x + 4}
\]
### Part 2: Finding \( (g \circ f)(x) \)
1. **Definition of Composite Function**: The composite function \( (g \circ f)(x) \) is defined as \( g(f(x)) \).
2. **Calculate \( f(x) \)**:
\[
f(x) = \sqrt{x}
\]
3. **Substitute \( f(x) \) into \( g(x) \)**:
\[
g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 4
\]
Thus, the expression for \( (g \circ f)(x) \) is:
\[
(g \circ f)(x) = \sqrt{x} + 4
\]
### Part 3: Finding \( (f \circ g)(5) \)
Substituting \( x = 5 \) into \( (f \circ g)(x) \):
\[
(f \circ g)(5) = \sqrt{5 + 4} = \sqrt{9} = 3
\]
### Part 4: Finding \( (g \circ f)(5) \)
Substituting \( x = 5 \) into \( (g \circ f)(x) \):
\[
(g \circ f)(5) = \sqrt{5} + 4
\]
### Summary of Results:
- a. \( (f \circ g)(x) = \sqrt{x + 4} \)
- b. \( (g \circ f)(x) = \sqrt{x} + 4 \)
- c. \( (f \circ g)(5) = 3 \)
- d. \( (g \circ f)(5) = \sqrt{5} + 4 \)
If you have any more questions or need further assistance, feel free to ask!
### Part 1: Finding \( (f \circ g)(x) \)
1. **Definition of Composite Function**: The composite function \( (f \circ g)(x) \) is defined as \( f(g(x)) \).
2. **Calculate \( g(x) \)**:
\[
g(x) = x + 4
\]
3. **Substitute \( g(x) \) into \( f(x) \)**:
\[
f(g(x)) = f(x + 4) = \sqrt{x + 4}
\]
Thus, the expression for \( (f \circ g)(x) \) is:
\[
(f \circ g)(x) = \sqrt{x + 4}
\]
### Part 2: Finding \( (g \circ f)(x) \)
1. **Definition of Composite Function**: The composite function \( (g \circ f)(x) \) is defined as \( g(f(x)) \).
2. **Calculate \( f(x) \)**:
\[
f(x) = \sqrt{x}
\]
3. **Substitute \( f(x) \) into \( g(x) \)**:
\[
g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 4
\]
Thus, the expression for \( (g \circ f)(x) \) is:
\[
(g \circ f)(x) = \sqrt{x} + 4
\]
### Part 3: Finding \( (f \circ g)(5) \)
Substituting \( x = 5 \) into \( (f \circ g)(x) \):
\[
(f \circ g)(5) = \sqrt{5 + 4} = \sqrt{9} = 3
\]
### Part 4: Finding \( (g \circ f)(5) \)
Substituting \( x = 5 \) into \( (g \circ f)(x) \):
\[
(g \circ f)(5) = \sqrt{5} + 4
\]
### Summary of Results:
- a. \( (f \circ g)(x) = \sqrt{x + 4} \)
- b. \( (g \circ f)(x) = \sqrt{x} + 4 \)
- c. \( (f \circ g)(5) = 3 \)
- d. \( (g \circ f)(5) = \sqrt{5} + 4 \)
If you have any more questions or need further assistance, feel free to ask!