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${:[f(x)=8x^(3)+5],[h(x)=root(3)(x-5)]:} Write (h@f)(x) as an expression in terms of x. (h@f)(x)=$

$\begin{array}{l} f(x)=8 x^{3}+5 \\ h(x)=\sqrt[3]{x-5} \end{array}$ $\newline$ Write $(h \circ f)(x)$ as an expression in terms of $x$ . $\newline$ $(h \circ f)(x)=$

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Domain and range of absolute value functions: equations

Full solution

Q.

\begin{array}{l} f(x)=8 x^{3}+5 \\ h(x)=\sqrt[3]{x-5} \end{array}

\newline

Write

(h \circ f)(x)

as an expression in terms of

x

.

\newline

(h \circ f)(x)=

Understand $h@f(x)$ : First, we need to understand that $(h@f)(x)$ means $h(f(x))$ . This means we will substitute the function $f(x)$ into the function $h(x)$ wherever there is an $x$ .
Substitute $f(x)$ in $h(x)$ : The function $f(x)$ is given as $f(x) = 8x^3 + 5$ . We will substitute this expression for $x$ in the function $h(x)$ .
Replace $x$ with $f(x)$ : The function $h(x)$ is given as $h(x) = \sqrt[3]{x - 5}$ . Now we will replace $x$ with $f(x)$ to get $h(f(x))$ . $\newline$ $h(f(x)) = \sqrt[3]{f(x) - 5}$
Substitute $f(x)$ in $h(f(x))$ : Substitute $f(x) = 8x^3 + 5$ into the expression for $h(f(x))$ : $\newline$ $h(f(x)) = \sqrt[3]{(8x^3 + 5) - 5}$
Simplify expression inside cube root: Simplify the expression inside the cube root: $\newline$ $h(f(x)) = \sqrt[3]{8x^3 + 5 - 5}$ $\newline$ $h(f(x)) = \sqrt[3]{8x^3}$
Take cube root of $8x^3$ : Now, we take the cube root of $8x^3$ :
$h(f(x)) = \sqrt[3]{8x^3}$
$h(f(x)) = 2x$

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