QuestionFind functions $f$ and $g$ where $(f \circ g)(x)=h(x)$ with $h(x)=\sqrt[3]{x^{2}-3}$ .

Studdy Solution

STEP 1

Assumptions1. The function $h(x)$ is given as $h(x)=\sqrt[3]{x^{}-3}$ . . We need to find two non-identity functions $f$ and $g$ such that the composition of $f$ and $g$ equals $h(x)$ , i.e., $(f \circ g)(x)=h(x)$ .

STEP 2

We can start by choosing a simple function for $g(x)$ . Let's choose $g(x)=x^{2}$ .
$g(x)=x^{2}$

STEP 3

Now, we need to find a function $f(x)$ such that when $g(x)$ is plugged into $f(x)$ , we get $h(x)$ . From the chosen function $g(x)$ , we have $x^{2}$ when $g(x)$ is plugged into $f(x)$ . So, we need a function $f(x)$ that can transform $x^{2}$ into $\sqrt[3]{x^{2}-3}$ .

STEP 4

Let's choose $f(x)=\sqrt[3]{x-3}$ . This function will take $x^{2}$ (from $g(x)$ ) and transform it into $\sqrt[3]{x^{2}-3}$ .
$f(x)=\sqrt[3]{x-3}$

STEP 5

Now, let's check if the composition of $f$ and $g$ equals $h(x)$ . The composition of $f$ and $g$ is written as $(f \circ g)(x)$ , which means $f(g(x))$ .
$f(g(x))=f(x^{2})=\sqrt[3]{(x^{2})-3}=\sqrt[3]{x^{2}-3}=h(x)$ So, the functions $f(x)=\sqrt[3]{x-3}$ and $g(x)=x^{2}$ satisfy the condition $(f \circ g)(x)=h(x)$ .
The solution is $(f(x), g(x))=(\sqrt[3]{x-3}, x^{2})$ .

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