Question

if $f(x) = x^2 - 2$ and $g(x) = x + 4$, find $(f \circ g)(3)$ and $(g \circ f)(-3)$.
a) $(f \circ g)(3)$
$(f \circ g)(3) = $
b) $(g \circ f)(-3)$
$(g \circ f)(-3) = $

Explanation:

Step1: Find \( (f \circ g)(3) \)

First, recall that \( (f \circ g)(x) = f(g(x)) \). So for \( x = 3 \), we first find \( g(3) \).
\( g(3) = 3 + 4 = 7 \)
Then substitute \( g(3) \) into \( f(x) \):
\( f(g(3)) = f(7) = 7^2 - 2 = 49 - 2 = 47 \)

Step2: Find \( (g \circ f)(-3) \)

Recall that \( (g \circ f)(x) = g(f(x)) \). So for \( x = -3 \), we first find \( f(-3) \).
\( f(-3) = (-3)^2 - 2 = 9 - 2 = 7 \)
Then substitute \( f(-3) \) into \( g(x) \):
\( g(f(-3)) = g(7) = 7 + 4 = 11 \)

Answer:

a) \( (f \circ g)(3) = 47 \)
b) \( (g \circ f)(-3) = 11 \)