Question

given (f(x) = 5x^2 + 2) and (g(x) = 2x^3)

((f \circ g)(x)) is a function of degree

((g \circ f)(x)) is a function of degree

Explanation:

⚡ Using what you learned: Evaluating Composite Functions

Step 1: Find the degree of \((f \circ g)(x)\)

Substitute \(g(x) = 2x^3\) into \(f(x) = 5x^2 + 2\):
\[ (f \circ g)(x) = f(g(x)) = 5(2x^3)^2 + 2 \]
\[ (f \circ g)(x) = 5(4x^6) + 2 = 20x^6 + 2 \]

The highest exponent of \(x\) is \(6\), so the degree is \(6\).

Step 2: Find the degree of \((g \circ f)(x)\)

Substitute \(f(x) = 5x^2 + 2\) into \(g(x) = 2x^3\):
\[ (g \circ f)(x) = g(f(x)) = 2(5x^2 + 2)^3 \]

Expanding \((5x^2 + 2)^3\) yields a leading term of:
\[ (5x^2)^3 = 125x^6 \]
\[ (g \circ f)(x) = 2(125x^6 + \dots) = 250x^6 + \dots \]

The highest exponent of \(x\) is \(6\), so the degree is \(6\).

Answer:

• \((f \circ g)(x)\) is a function of degree 6
• \((g \circ f)(x)\) is a function of degree 6