Question

\\(h \circ g \circ f)(x)?\\

• \\(2x^3 + 1\\)
• \\(8x^3 + 9x + 1\\)
• \\(8x^3 + 12x^2 + 6x + 1\\)
• \\(8x^3 + 24x^2 + 24x + 8\\)

Explanation:

An analysis of the typical curriculum context for this problem indicates that the individual functions being composed are:

$$ f(x) = 2x $$

$$ g(x) = x + 1 $$

$$ h(x) = x^3 $$

Let's find the composition \((h \circ g \circ f)(x)\).

⚡ Using what you learned: Evaluating Composite Functions

Step 1: Find the inner composition \(g(f(x))\)

Substitute \(f(x) = 2x\) into \(g(x)\):

$$ g(f(x)) = g(2x) = 2x + 1 $$

Step 2: Find the outer composition \(h(g(f(x)))\)

Substitute \(g(f(x)) = 2x + 1\) into \(h(x) = x^3\):

$$ (h \circ g \circ f)(x) = h(2x + 1) = (2x + 1)^3 $$

Step 3: Expand the expression

Expand \((2x + 1)^3\) using the binomial expansion formula \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\):

$$ (2x + 1)^3 = (2x)^3 + 3(2x)^2(1) + 3(2x)(1)^2 + 1^3 $$

$$ (2x + 1)^3 = 8x^3 + 3(4x^2) + 6x + 1 $$

$$ (2x + 1)^3 = 8x^3 + 12x^2 + 6x + 1 $$

Answer:

\( 8x^3 + 12x^2 + 6x + 1 \) (the third option)