Question

for $f(x)=x^{2}+9$ and $g(x)=x^{2}-9$, find the following functions.
a. $(f\circ g)(x);$ b. $(g\circ f)(x);$ c. $(f\circ g)(4);$ d. $(g\circ f)(4)$
a. $(f\circ g)(x)=\square$
(simplify your answer.)
b. $(g\circ f)(x)=\square$
(simplify your answer.)
c. $(f\circ g)(4)=\square$
d. $(g\circ f)(4)=\square$

Explanation:

Step1: Define composite function (f∘g)(x)

$(f \circ g)(x) = f(g(x))$
Substitute $g(x)=x^2-9$ into $f(x)$:
$f(g(x)) = (x^2 - 9)^2 + 9$
Expand and simplify:
$(x^2 - 9)^2 + 9 = x^4 - 18x^2 + 81 + 9 = x^4 - 18x^2 + 90$

Step2: Define composite function (g∘f)(x)

$(g \circ f)(x) = g(f(x))$
Substitute $f(x)=x^2+9$ into $g(x)$:
$g(f(x)) = (x^2 + 9)^2 - 9$
Expand and simplify:
$(x^2 + 9)^2 - 9 = x^4 + 18x^2 + 81 - 9 = x^4 + 18x^2 + 72$

Step3: Calculate (f∘g)(4)

Substitute $x=4$ into $(f \circ g)(x)$:
$(f \circ g)(4) = 4^4 - 18(4)^2 + 90$
Calculate each term:
$4^4=256$, $18(4)^2=18\times16=288$
$256 - 288 + 90 = 58$

Step4: Calculate (g∘f)(4)

Substitute $x=4$ into $(g \circ f)(x)$:
$(g \circ f)(4) = 4^4 + 18(4)^2 + 72$
Calculate each term:
$4^4=256$, $18(4)^2=18\times16=288$
$256 + 288 + 72 = 616$

Answer:

a. $\boldsymbol{x^4 - 18x^2 + 90}$
b. $\boldsymbol{x^4 + 18x^2 + 72}$
c. $\boldsymbol{58}$
d. $\boldsymbol{616}$