SOLUTION: Use the following to answer questions f(x)=x^2+x and g(x)=x-5 Find h(x)=(f°g)(x) State the domain of h(x)=(f°g)(x) Find h(x)=(g°f)(x) State the domain of h(x)=(g°f)(x)

Hi, there--

THE PROBLEM:
Given that  f(x)=x^2+x and g(x)=x-5 

Part 1:
Find h(x)=(f°g)(x)
State the domain of h(x)=(f°g)(x)

Part 2:
Find h(x)=(g°f)(x)
State the domain of h(x)=(g°f)(x)

A SOLUTION:
The function h(x) = (f°g)(x) is called the composite function. We sometimes see it written
h(x) = f(g(x))

Let's start with the basics. The function f is defined as
f(x) = x^2 + x

We know that we can find the value of the function for any x in the domain by substitution. 
For example, to find f(3), I substitute 3 for x in the function.
f(3) = (3)^2 + (3) 
f(3) = 9 + 3
f(3) = 12

In a composite function, we substitute the expression for the second function for x in the 
first function. In this case, g(x) = x-5. To find f°g, I substitute x-5 for x in the function f.
f(x) = x^2 + x
f(g(x)) = (x-5)^2 + (x-5)

We can stop here, or we can simplify this if we want.
f(g(x) = x^2 - 10x +25 + x - 5
f(g(x)) = x^2 - 9x + 20
h(x) = f(g(x) = x^2 - 9x + 20

The domain of f(x) is all real numbers because we can square any real number and 
add it to itself and still have a real number.
The domain of g(x) is all real numbers because we can subtract 5 from any real number 
and still have a real number.

Similarly, the domain of h(x) is all real numbers.

Part 2:
You can find h(x) = (g°f)(x) by using the same process as in Part 1. Feel free to email me if 
you have questions or get stuck.

Mrs. Figgy
math.in.the.vortex@gmail.com