Question

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Explanation:

Identify the given functions

We are given two linear functions:

$$ f(x) = 3x + 2 \quad \text{and} \quad g(x) = 4x $$

Our goal is to find the composite functions \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and state the domain of each.

Find the composite function (f ∘ g) and its domain

Using the Composite Functions knowledge point:

$$ (f \circ g)(x) = f(g(x)) = f(4x) = 3(4x) + 2 = 12x + 2 $$

Using the Linear Function Domain and Interval Notation knowledge points:
Since \(g(x)\) and \(f(g(x))\) are linear polynomials, there are no restrictions.
The domain is \((-\infty, \infty)\).

Find the composite function (g ∘ f) and its domain

Using the Composite Functions knowledge point:

$$ (g \circ f)(x) = g(f(x)) = g(3x + 2) = 4(3x + 2) = 12x + 8 $$

Using the Linear Function Domain and Interval Notation knowledge points:
Since \(f(x)\) and \(g(f(x))\) are linear polynomials, there are no restrictions.
The domain is \((-\infty, \infty)\).

Find the composite function (f ∘ f) and its domain

Using the Composite Functions knowledge point:

$$ (f \circ f)(x) = f(f(x)) = f(3x + 2) = 3(3x + 2) + 2 = 9x + 6 + 2 = 9x + 8 $$

Using the Linear Function Domain and Interval Notation knowledge points:
Since \(f(x)\) and \(f(f(x))\) are linear polynomials, there are no restrictions.
The domain is \((-\infty, \infty)\).

Find the composite function (g ∘ g) and its domain

Using the Composite Functions knowledge point:

$$ (g \circ g)(x) = g(g(x)) = g(4x) = 4(4x) = 16x $$

Using the Linear Function Domain and Interval Notation knowledge points:
Since \(g(x)\) and \(g(g(x))\) are linear polynomials, there are no restrictions.
The domain is \((-\infty, \infty)\).

Answer:

Question 1

• Composite function: \((f \circ g)(x) = 12x + 2\)
• Domain: \((-\infty, \infty)\)

Question 2

• Composite function: \((g \circ f)(x) = 12x + 8\)
• Domain: \((-\infty, \infty)\)

Question 3

• Composite function: \((f \circ f)(x) = 9x + 8\)
• Domain: \((-\infty, \infty)\)

Question 4

• Composite function: \((g \circ g)(x) = 16x\)
• Domain: \((-\infty, \infty)\)