Question

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Explanation:

Define the composite function

To find the expression equivalent to \((m \circ n)(x)\), we must evaluate the composite function \(m(n(x))\). This means we substitute the entire function \(n(x)\) into every instance of \(x\) in the function \(m(x)\).
Given:

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

$$n(x) = 5x + 9$$

Substitute n(x) into m(x)

Substitute the expression for \(n(x)\) into \(m(x)\):

$$(m \circ n)(x) = m(n(x)) = (5x + 9)^2 + 3$$

Expand the squared binomial

Expand the term \((5x + 9)^2\) using the perfect square trinomial formula \((a + b)^2 = a^2 + 2ab + b^2\):

$$(5x + 9)^2 = (5x)^2 + 2(5x)(9) + 9^2$$

$$(5x + 9)^2 = 25x^2 + 90x + 81$$

Simplify the expression

Add the remaining constant term \(3\) to the expanded expression:

$$(m \circ n)(x) = 25x^2 + 90x + 81 + 3$$

$$(m \circ n)(x) = 25x^2 + 90x + 84$$

Answer:

• (A) \(5x^3 + 9x^2 + 15x + 27\)
• (B) \(25x^2 + 90x + 84\) (Correct answer)
• (C) \(x^2 + 5x + 12\)
• (D) \(5x^2 + 24\)