Question

use $f(x)=5x - 3$ and $g(x)=2 - x^{2}$ to evaluate the expression.
(a) $(fcirc f)(x)$
(b) $(gcirc g)(x)$

Explanation:

Step1: Recall composition of functions

The composition $(f\circ f)(x)=f(f(x))$. Given $f(x) = 5x - 3$, we substitute $f(x)$ into itself.

Step2: Substitute $f(x)$ into $f$

$f(f(x))=5(5x - 3)-3$.

Step3: Expand the expression

$5(5x - 3)-3=25x-15 - 3=25x-18$.

Step4: Recall composition of functions for $g$

The composition $(g\circ g)(x)=g(g(x))$. Given $g(x)=2 - x^{2}$, we substitute $g(x)$ into itself.

Step5: Substitute $g(x)$ into $g$

$g(g(x))=2-(2 - x^{2})^{2}$.

Step6: Expand $(2 - x^{2})^{2}$

$(2 - x^{2})^{2}=4-4x^{2}+x^{4}$ according to $(a - b)^{2}=a^{2}-2ab + b^{2}$ where $a = 2$ and $b=x^{2}$.

Step7: Simplify $g(g(x))$

$g(g(x))=2-(4 - 4x^{2}+x^{4})=2-4 + 4x^{2}-x^{4}=-x^{4}+4x^{2}-2$.

Answer:

(a) $25x - 18$
(b) $-x^{4}+4x^{2}-2$