Question

given that $f(x) = -4x - 9$ and $g(x) = -2x - 4$, determine each of the following. make sure to fully simplify your answer.
(a) $(f \circ g)(x)=$
(b) $(g \circ f)(x)=$
(c) $(f \circ f)(x)=$
(d) $(g \circ g)(x)=$
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Explanation:

Part (a): \((f \circ g)(x)\)

Step1: Recall composition definition

\((f \circ g)(x) = f(g(x))\). So substitute \(g(x)\) into \(f\).
\(g(x) = -2x - 4\), so \(f(g(x)) = f(-2x - 4)\).

Step2: Substitute into \(f(x)\)

\(f(x) = -4x - 9\), replace \(x\) with \(-2x - 4\):
\(f(-2x - 4) = -4(-2x - 4) - 9\)

Step3: Simplify the expression

First, distribute \(-4\): \(-4(-2x) + (-4)(-4) - 9 = 8x + 16 - 9\)
Then combine like terms: \(8x + 7\)

Step1: Recall composition definition

\((g \circ f)(x) = g(f(x))\). Substitute \(f(x)\) into \(g\).
\(f(x) = -4x - 9\), so \(g(f(x)) = g(-4x - 9)\).

Step2: Substitute into \(g(x)\)

\(g(x) = -2x - 4\), replace \(x\) with \(-4x - 9\):
\(g(-4x - 9) = -2(-4x - 9) - 4\)

Step3: Simplify the expression

Distribute \(-2\): \(-2(-4x) + (-2)(-9) - 4 = 8x + 18 - 4\)
Combine like terms: \(8x + 14\)

Step1: Recall composition definition

\((f \circ f)(x) = f(f(x))\). Substitute \(f(x)\) into \(f\).
\(f(x) = -4x - 9\), so \(f(f(x)) = f(-4x - 9)\).

Step2: Substitute into \(f(x)\)

\(f(x) = -4x - 9\), replace \(x\) with \(-4x - 9\):
\(f(-4x - 9) = -4(-4x - 9) - 9\)

Step3: Simplify the expression

Distribute \(-4\): \(-4(-4x) + (-4)(-9) - 9 = 16x + 36 - 9\)
Combine like terms: \(16x + 27\)

Answer:

\(8x + 7\)

Part (b): \((g \circ f)(x)\)