Question

composites - 2
solve each of the following using the given functions.
f(x)=3x + 2
$g(x)=\frac{x}{3}+5$
h(x)=x²

1. f(g(x))=
2. g(h(x))=
3. f(h(x))=
4. f(h(0))=
5. h(f(2))=
6. (f.h)(-2)=

Explanation:

Step1: Find f(g(x))

Substitute g(x) into f(x). Given f(x)=3x + 2 and g(x)=\frac{x}{3}+5, then f(g(x))=3(\frac{x}{3}+5)+2.
First, distribute the 3: 3\times\frac{x}{3}+3\times5 + 2=x + 15+2=x + 17.

Step2: Find g(h(x))

Substitute h(x) into g(x). Given g(x)=\frac{x}{3}+5 and h(x)=x^{2}, then g(h(x))=\frac{x^{2}}{3}+5.

Step3: Find f(h(x))

Substitute h(x) into f(x). Given f(x)=3x + 2 and h(x)=x^{2}, then f(h(x))=3x^{2}+2.

Step4: Find f(h(0))

First, find h(0). Since h(x)=x^{2}, then h(0)=0^{2}=0.
Then substitute h(0) into f(x). Since f(x)=3x + 2, then f(h(0))=f(0)=3\times0+2=2.

Step5: Find h(f(2))

First, find f(2). Since f(x)=3x + 2, then f(2)=3\times2+2=6 + 2=8.
Then substitute f(2) into h(x). Since h(x)=x^{2}, then h(f(2))=h(8)=8^{2}=64.

Step6: Find (f\cdot h)(-2)

(f\cdot h)(x)=f(x)\cdot h(x)=(3x + 2)\cdot x^{2}=3x^{3}+2x^{2}.
Substitute x = - 2 into (f\cdot h)(x): (f\cdot h)(-2)=3\times(-2)^{3}+2\times(-2)^{2}.
Calculate (-2)^{3}=-8 and (-2)^{2}=4. Then (f\cdot h)(-2)=3\times(-8)+2\times4=-24 + 8=-16.

Answer:

1. f(g(x))=x + 17
2. g(h(x))=\frac{x^{2}}{3}+5
3. f(h(x))=3x^{2}+2
4. f(h(0))=2
5. h(f(2))=64
6. (f\cdot h)(-2)=-16