Question

homework 16: section 3.4
score: 77.5/160 answered: 8/16
question 8
given that $f(x)=x^2 - 5x$ and $g(x)=x + 6$, calculate
(a) $(f\circ g)(2)=$
(b) $(g\circ f)(2)=$
question help: video message instructor

Explanation:

Part (a)

Step1: Recall function composition

To find \((f \circ g)(2)\), we first find \(g(2)\), then substitute that result into \(f(x)\).

Step2: Calculate \(g(2)\)

Given \(g(x) = x + 6\), substitute \(x = 2\):
\(g(2)=2 + 6 = 8\)

Step3: Substitute \(g(2)\) into \(f(x)\)

Given \(f(x)=x^{2}-5x\), substitute \(x = 8\) (since \(g(2) = 8\)):
\(f(8)=8^{2}-5\times8 = 64 - 40 = 24\)

Part (b)

Step1: Recall function composition

To find \((g \circ f)(2)\), we first find \(f(2)\), then substitute that result into \(g(x)\).

Step2: Calculate \(f(2)\)

Given \(f(x)=x^{2}-5x\), substitute \(x = 2\):
\(f(2)=2^{2}-5\times2 = 4 - 10 = -6\)

Step3: Substitute \(f(2)\) into \(g(x)\)

Given \(g(x)=x + 6\), substitute \(x = -6\) (since \(f(2)= -6\)):
\(g(-6)=-6 + 6 = 0\)

Answer:

(a) \(\boldsymbol{24}\)
(b) \(\boldsymbol{0}\)