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.)

Explanation:

Find the composition (f o g)(x)

Using the Function Composition knowledge point

$$ (f \circ g)(x) = f(g(x)) = f(x + 6) = \sqrt{x + 6} $$

Find the composition (g o f)(x)

Using the Function Composition knowledge point

$$ (g \circ f)(x) = g(f(x)) = g(\sqrt{x}) = \sqrt{x} + 6 $$

Evaluate (f o g)(3)

Using the Function Composition knowledge point

$$ (f \circ g)(3) = \sqrt{3 + 6} = \sqrt{9} = 3 $$

Evaluate (g o f)(3)

Using the Function Composition knowledge point

$$ (g \circ f)(3) = \sqrt{3} + 6 $$

Answer:

Question a

\( (f \circ g)(x) = \) <blank>\(\sqrt{x + 6}\)</blank>

Question b

\( (g \circ f)(x) = \) <blank>\(\sqrt{x} + 6\)</blank>

Question c

\( (f \circ g)(3) = \) <blank>\(3\)</blank>

Question d

\( (g \circ f)(3) = \) <blank>\(\sqrt{3} + 6\)</blank>