The functions mapping R into R are defined as : \(f\left(x \right)=x^{3} - 4x,…
2017
The functions mapping R into R are defined as :
\(f\left(x \right)=x^{3} - 4x, g\left(x \right)=\frac{1}{x^{2}+1}\) ) h(x)=x4
Then find the value of the following composite functions :
\(h_{o}g\left(x \right)\) and \(h_{o}g_{o}f\left(x \right)\)
- A.
\(\left ( x^{2}+1 \right )^{4}\)and\(\left [ \left ( x^{3}-4x \right )^{2}+1 \right ]^{4}\) - B.
\(\left ( x^{2}+1 \right )^{4}\)and\(\left [ \left ( x^{3}-4x \right )^{2}+1 \right ]^{-4}\) - C.
\(\left ( x^{2}+1 \right )^{-4}\)and\(\left [ \left ( x^{2}-4x \right )^{2}+1 \right ]^{4}\) - D.
\(\left ( x^{2}+1 \right )^{-4}\)and\(\left [ \left ( x^{3}-4x \right )^{2}+1 \right ]^{-4}\)
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Correct answer: D
Given: f(x) = x^3 - 4x, g(x) = 1/(x^2 + 1), and h(x) = x^4.
Compute h ∘ g:
h(g(x)) = (g(x))^4 = (1/(x^2 + 1))^4 = 1/(x^2 + 1)^4 = (x^2 + 1)^-4.
Compute h ∘ g ∘ f:
h(g(f(x))) = (g(f(x)))^4 = (1/((f(x))^2 + 1))^4 = 1/((x^3 - 4x)^2 + 1)^4 = [((x^3 - 4x)^2 + 1)]^-4.
Final answers:
h ∘ g (x) = (x^2 + 1)^-4.
h ∘ g ∘ f (x) = [((x^3 - 4x)^2 + 1)]^-4.
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