Tiffany C. answered • 08/27/20

Experienced Instructor with Math and Database Expertise

Hi Tamese,

Your question is an example of a composition of functions,which is pronounced f-compose-*g* of *x.*

The way to think about these types of questions is that you can replace the x in the F(x) function with the expression or value of g(x).

Also, another way to write this problem is:

(* f *o *g*)(*x*) = f(g(x))

**Let's get to solving!**

So f(x)=x^2+1 and g(x)=x-4 and we want to solve for f(g(10).

There are two ways to think about solving:

__Option1:__

"Working inside out"

First, solve for the innermost functions value and plug it into the function that wraps it. In our case solve for g(x) where x=10 then solve for f(x) with x =g(10).

g(10)= 10-4

g(10)=6

Next we solve for f(g(10)), since g(10)=6 we can rewrite the function as f(6).

f(6)= 6^2+1

f(6)=36+1

f(6)=37

**Therefore ****( f o g)(x) =f(g(10)) = 37.**

__Option 2:__

"Write A New Equation"

In this option we will be writing out what f-compose-*g* of x is an equation and plugging 10 into our new expression.

For this we will be replacing all x's in f(x) with the function of g

So

(* f *o *g*)(*x*) = (x-4)^2+1

Now we solve for where x=10

(* f *o *g*)(10)= (10-4)^2+1

(* f *o *g*)(10)= (6)^2 +1

(* f *o *g*)(10)= 36+1

(* f *o *g*)(10)=37

**Therefore ****( f o g)(x) =f(g(10)) = 37.**

__In conclusion__

**( f o g)(x) =f(g(10)) = 37**

Either method will get you to your answer, use whichever works best for you.