To express the function [tex]\( h(x) = \sqrt[3]{5 x^2 - 3 x} \)[/tex] as a composition of two functions, [tex]\( f \)[/tex] and [tex]\( g \)[/tex], where [tex]\( (f \circ g)(x) = h(x) \)[/tex], we need to identify suitable definitions for [tex]\( f \)[/tex] and [tex]\( g \)[/tex].
1. Define the inner function [tex]\( g(x) \)[/tex]:
We start by examining the expression inside the cube root of [tex]\( h(x) \)[/tex]. Let:
[tex]\[
g(x) = 5 x^2 - 3 x
\][/tex]
This function [tex]\( g \)[/tex] takes the input [tex]\( x \)[/tex] and maps it to [tex]\( 5 x^2 - 3 x \)[/tex].
2. Define the outer function [tex]\( f(u) \)[/tex]:
Next, we need a function [tex]\( f \)[/tex] that, when applied to the result of [tex]\( g(x) \)[/tex], gives us [tex]\( h(x) \)[/tex]. Since [tex]\( h(x) \)[/tex] is a cube root applied to [tex]\( g(x) \)[/tex], define:
[tex]\[
f(u) = \sqrt[3]{u}
\][/tex]
This function takes the input [tex]\( u \)[/tex] and maps it to [tex]\( u^{1/3} \)[/tex].
3. Compose the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
Now, we compose [tex]\( f \)[/tex] and [tex]\( g \)[/tex] to get [tex]\( h \)[/tex]. The composition [tex]\( (f \circ g)(x) \)[/tex] means we first apply [tex]\( g \)[/tex] to [tex]\( x \)[/tex] and then apply [tex]\( f \)[/tex] to the result:
[tex]\[
h(x) = (f \circ g)(x) = f(g(x))
\][/tex]
Substitute [tex]\( g(x) = 5 x^2 - 3 x \)[/tex] into [tex]\( f(u) \)[/tex]:
[tex]\[
h(x) = f(5 x^2 - 3 x) = \sqrt[3]{5 x^2 - 3 x}
\][/tex]
In summary, the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] that achieve [tex]\( (f \circ g)(x) = h(x) \)[/tex] are:
[tex]\[
g(x) = 5 x^2 - 3 x
\][/tex]
[tex]\[
f(u) = \sqrt[3]{u}
\][/tex]
Therefore, [tex]\( h(x) \)[/tex] can be expressed as the composition [tex]\( h(x) = f(g(x)) = \sqrt[3]{5 x^2 - 3 x} \)[/tex], where:
[tex]\[
g(x) = 5 x^2 - 3 x
\][/tex]
[tex]\[
f(u) = \sqrt[3]{u}
\][/tex]
