Question

question list
for $f(x)=x^2 + 4$ and $g(x)=x^2 - 8$, find the following functions.
a. $(f \circ g)(x)$; b. $(g \circ f)(x)$; c. $(f \circ g)(4)$; d. $(g \circ f)(4)$
question 2
a. $(f \circ g)(x)=\square$
(simplify your answer.)
b. $(g \circ f)(x)=\square$
(simplify your answer.)
question 3
c. $(f \circ g)(4)=\square$
question 4
d. $(g \circ f)(4)=\square$
question 5

Explanation:

Step1: Define composite $(f\circ g)(x)$

Substitute $g(x)$ into $f(x)$:
$f(g(x)) = (g(x))^2 + 4$
Substitute $g(x)=x^2-8$:
$f(g(x)) = (x^2-8)^2 + 4$
Expand and simplify:
$(x^2-8)^2 + 4 = x^4 -16x^2 +64 +4 = x^4 -16x^2 +68$

Step2: Define composite $(g\circ f)(x)$

Substitute $f(x)$ into $g(x)$:
$g(f(x)) = (f(x))^2 -8$
Substitute $f(x)=x^2+4$:
$g(f(x)) = (x^2+4)^2 -8$
Expand and simplify:
$(x^2+4)^2 -8 = x^4 +8x^2 +16 -8 = x^4 +8x^2 +8$

Step3: Calculate $(f\circ g)(4)$

Use $(f\circ g)(x)=x^4 -16x^2 +68$, substitute $x=4$:
$(4)^4 -16(4)^2 +68 = 256 -16(16) +68 = 256 -256 +68 = 68$

Step4: Calculate $(g\circ f)(4)$

Use $(g\circ f)(x)=x^4 +8x^2 +8$, substitute $x=4$:
$(4)^4 +8(4)^2 +8 = 256 +8(16) +8 = 256 +128 +8 = 392$

Answer:

a. $x^4 -16x^2 +68$
b. $x^4 +8x^2 +8$
c. $68$
d. $392$