Question
For \( f(x)=\sqrt{x} \) and \( g(x)=x+6 \), find the following functions. a. \( (f \circ g)(x) ; b .(g \circ f)(x) \); c. \( (f \circ g)(3) \); d. \( (g \circ f)(3) \) a. \( (f \circ g)(x)=\square \) (Simplify your answer.) b. \( (g \circ f)(x)=\square \) \( ( \) Simplify your answer. \( ) \) c. \( (f \circ g)(3)=\square \) (Simplify your answer.) d. \( (g \circ f)(3)=\square \) (Simplify your answer.)
Ask by Carroll Mccoy. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
a. \( (f \circ g)(x) = \sqrt{x + 6} \)
b. \( (g \circ f)(x) = \sqrt{x} + 6 \)
c. \( (f \circ g)(3) = 3 \)
d. \( (g \circ f)(3) = \sqrt{3} + 6 \) (approximately 7.732)
Solution
Let's evaluate each part step by step using the given functions \( f(x) = \sqrt{x} \) and \( g(x) = x + 6 \).
### a. \( (f \circ g)(x) \)
\[
(f \circ g)(x) = f(g(x)) = f(x + 6) = \sqrt{x + 6}
\]
**Answer:** \( (f \circ g)(x) = \sqrt{x + 6} \)
### b. \( (g \circ f)(x) \)
\[
(g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 6
\]
**Answer:** \( (g \circ f)(x) = \sqrt{x} + 6 \)
### c. \( (f \circ g)(3) \)
\[
(f \circ g)(3) = \sqrt{3 + 6} = \sqrt{9} = 3
\]
**Answer:** \( (f \circ g)(3) = 3 \)
### d. \( (g \circ f)(3) \)
\[
(g \circ f)(3) = \sqrt{3} + 6 \approx 1.732 + 6 = 7.732
\]
**Answer:** \( (g \circ f)(3) = \sqrt{3} + 6 \) (approximately 7.732)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find the required functions, let's break them down step by step: a. \( (f \circ g)(x) = f(g(x)) = f(x + 6) = \sqrt{x + 6} \) b. \( (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 6 \) c. \( (f \circ g)(3) = \sqrt{3 + 6} = \sqrt{9} = 3 \) d. \( (g \circ f)(3) = \sqrt{3} + 6 \approx 1.732 + 6 = 7.732 \) So, the final answers are: a. \( (f \circ g)(x) = \sqrt{x + 6} \) b. \( (g \circ f)(x) = \sqrt{x} + 6 \) c. \( (f \circ g)(3) = 3 \) d. \( (g \circ f)(3) = \sqrt{3} + 6 \)
