Question

which expression represents the composition \\(g \circ f \circ h(x)\\) for the functions below?

\\(f(x) = 5x - 4\\)
\\(g(x) = 5x^3\\)
\\(h(x) = 3x\\)

\\(\bigcirc\\) \\(5(15x - 4)^3\\)
\\(\bigcirc\\) \\((75x - 20)^3\\)
\\(\bigcirc\\) \\(5(5x - 4)^3\\)
\\(\bigcirc\\) \\(675x^3 - 4\\)

Explanation:

Evaluate the inner composition

We first find the inner composite function \(f(h(x))\).

$$ h(x) = 3x $$

$$ f(h(x)) = f(3x) = 5(3x) - 4 = 15x - 4 $$

Apply the outer function

We now substitute \(f(h(x))\) into the outer function \(g(x)\).

$$ g(x) = 5x^3 $$

$$ [g \circ f \circ h](x) = g(f(h(x))) = g(15x - 4) $$

$$ g(15x - 4) = 5(15x - 4)^3 $$

Answer:

• (A) \(5(15x - 4)^3\) (Correct answer)
• (B) \((75x - 20)^3\)
• (C) \(5(5x - 4)^3\)
• (D) \(675x^3 - 4\)