Answer:
[tex]h(g(x))=x^2+x-2[/tex]
[tex]h(g(f(x))= x^2-19x+88[/tex]
Step-by-step explanation:
Composite Function
Suppose f(x) and g(x) are real functions, the composite function named [tex](f\circ g)(x)[/tex] is defined as:
[tex](f\circ g)(x)=f(g(x))[/tex]
The composite function can be found by substituting g into f.
We are given these functions:
[tex]f(x)=9-x[/tex]
[tex]g(x)=x^2+x[/tex]
[tex]h(x)=x-2[/tex]
Find:
a) h(g(x))
Substituting g into h:
[tex]h(g(x))=x^2+x-2[/tex]
b) h(g(f(x))
First we find g(f(x)):
[tex]g(f(x))=(9-x)^2+(9-x)[/tex]
Operating:
[tex]g(f(x))=81-18x+x^2+9-x[/tex]
Simplifying:
[tex]g(f(x))=90-19x+x^2[/tex]
Now find h(g(f(x)) replacing the above equation into h:
[tex]h(g(f(x))= 90-19x+x^2-2[/tex]
Simplifying and rearranging:
[tex]h(g(f(x))= 88-19x+x^2[/tex]
[tex]h(g(f(x))= x^2-19x+88[/tex]
