Question

for f(x)=2x - 5 and g(x)=x^2 - 2, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(2); d. (g ∘ f)(2) a. (f ∘ g)(x)= (simplify your answer.) b. (g ∘ f)(x)= (simplify your answer.) c. (f ∘ g)(2)= (simplify your answer.) d. (g ∘ f)(2)= (simplify your answer.)

Explanation:

Step1: Recall composition formula

$(f\circ g)(x)=f(g(x))$ and $(g\circ f)(x)=g(f(x))$.

Step2: Find $(f\circ g)(x)$

Substitute $g(x)=x^{2}-2$ into $f(x)$. So $f(g(x)) = 2(x^{2}-2)-5$. Then simplify:
\[

$$\begin{align*} f(g(x))&=2x^{2}-4 - 5\\ &=2x^{2}-9 \end{align*}$$

\]

Step3: Find $(g\circ f)(x)$

Substitute $f(x)=2x - 5$ into $g(x)$. So $g(f(x))=(2x - 5)^{2}-2$. Expand $(2x - 5)^{2}$ using $(a - b)^{2}=a^{2}-2ab + b^{2}$ where $a = 2x$ and $b = 5$.
\[

$$\begin{align*} g(f(x))&=(2x)^{2}-2\times(2x)\times5+5^{2}-2\\ &=4x^{2}-20x + 25-2\\ &=4x^{2}-20x+23 \end{align*}$$

\]

Step4: Find $(f\circ g)(2)$

Substitute $x = 2$ into $(f\circ g)(x)=2x^{2}-9$. So $(f\circ g)(2)=2\times2^{2}-9=2\times4 - 9=8 - 9=-1$.

Step5: Find $(g\circ f)(2)$

Substitute $x = 2$ into $(g\circ f)(x)=4x^{2}-20x + 23$. So $(g\circ f)(2)=4\times2^{2}-20\times2+23=4\times4-40 + 23=16-40 + 23=-1$.

Answer:

a. $(f\circ g)(x)=2x^{2}-9$
b. $(g\circ f)(x)=4x^{2}-20x + 23$
c. $(f\circ g)(2)=-1$
d. $(g\circ f)(2)=-1$