Question

II. Let f(x) = 9 – x , g(x) = x2 + x, and h(x) = x – 2. Compute the following:

Answer 1

Answer:

$h(g(x))=x^2+x-2$

$h(g(f(x))= x^2-19x+88$

Step-by-step explanation:

Composite Function

Suppose f(x) and g(x) are real functions, the composite function named $(f\circ g)(x)$ is defined as:

$(f\circ g)(x)=f(g(x))$

The composite function can be found by substituting g into f.

We are given these functions:

$f(x)=9-x$

$g(x)=x^2+x$

$h(x)=x-2$

Find:

a) h(g(x))

Substituting g into h:

$h(g(x))=x^2+x-2$

b) h(g(f(x))

First we find g(f(x)):

$g(f(x))=(9-x)^2+(9-x)$

Operating:

$g(f(x))=81-18x+x^2+9-x$

Simplifying:

$g(f(x))=90-19x+x^2$

Now find h(g(f(x)) replacing the above equation into h:

$h(g(f(x))= 90-19x+x^2-2$

Simplifying and rearranging:

$h(g(f(x))= 88-19x+x^2$

$h(g(f(x))= x^2-19x+88$