Question

homework 16: section 3.4
score: 67.5/160 answered: 7/16
question 7
given that $f(x)=x^2 - 3x$ and $g(x)=x + 9$, calculate
(a) $(f\circ g)(x)=$
(b) $(g\circ f)(x)=$
(c) $(f\circ f)(x)=$
(d) $(g\circ g)(x)=$

Explanation:

Part (a)

Step1: Recall composition definition

The composition \((f \circ g)(x)\) means \(f(g(x))\). So we substitute \(g(x)\) into \(f(x)\).
Given \(f(x)=x^2 - 3x\) and \(g(x)=x + 9\), we replace \(x\) in \(f(x)\) with \(g(x)=x + 9\).

Step2: Substitute and simplify

$$
LATEXBLOCK0
$$

Step1: Recall composition definition

The composition \((g \circ f)(x)\) means \(g(f(x))\). So we substitute \(f(x)\) into \(g(x)\).
Given \(g(x)=x + 9\) and \(f(x)=x^2-3x\), we replace \(x\) in \(g(x)\) with \(f(x)=x^2-3x\).

Step2: Substitute and simplify

$$
LATEXBLOCK0
$$

(We just substitute \(x^2 - 3x\) for \(x\) in \(g(x)=x + 9\), so we get \(x^2-3x + 9\))

Step1: Recall composition definition

The composition \((f \circ f)(x)\) means \(f(f(x))\). So we substitute \(f(x)\) into \(f(x)\) again.
Given \(f(x)=x^2-3x\), we replace \(x\) in \(f(x)\) with \(f(x)=x^2-3x\).

Step2: Substitute and simplify

$$
LATEXBLOCK0
$$

Answer:

\(x^2 + 15x + 54\)

Part (b)