Question

for f(x)=x² + 9 and g(x)=x² - 3, find the following functions. a. (f ∘ g)(x); b. (g ∘ f)(x); c. (f ∘ g)(4); d. (g ∘ f)(4) a. (f ∘ g)(x)=x⁴ - 6x² + 18 (simplify your answer.) b. (g ∘ f)(x)=x⁴ + 18x² + 78 (simplify your answer.) c. (f ∘ g)(4)=178 d. (g ∘ f)(4)=□

Explanation:

Step1: Recall composition formula

$(g\circ f)(x)=g(f(x))$. We know $f(x)=x^{2}+9$ and $g(x)=x^{2}-3$.

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

$g(f(x))=(x^{2}+9)^{2}-3$.

Step3: Expand $(x^{2}+9)^{2}$

Using $(a + b)^{2}=a^{2}+2ab + b^{2}$ where $a = x^{2}$ and $b = 9$, we get $(x^{2}+9)^{2}=(x^{2})^{2}+2\times x^{2}\times9 + 9^{2}=x^{4}+18x^{2}+81$.

Step4: Calculate $g(f(x))$

$g(f(x))=x^{4}+18x^{2}+81 - 3=x^{4}+18x^{2}+78$.

Step5: Find $(g\circ f)(4)$

Substitute $x = 4$ into $g(f(x))$. So $(g\circ f)(4)=4^{4}+18\times4^{2}+78$.

Step6: Calculate powers and multiplications

$4^{4}=256$, $18\times4^{2}=18\times16 = 288$.

Step7: Calculate the sum

$(g\circ f)(4)=256+288+78=622$.

Answer:

622