1.4: Arithmetic Combinations...Continued

The first thing we learned today is how to find f composed of g of x:

For example:

Findwhen...

Step #1

The first step would be to rewrite as

Step #2

Next you find g(3) by plugging in 3 for every x.

Step #3

Now you use g(3)=14 to find f(g(3)).  Substitute 14 for g(3) and rewrite the equation as f(14) and solve.

Answer: 

Is it commutative?

No (except for a few examples that we will learn later)

We can prove this by...

Solving  and comparing our answer to 

Step #1
The first step would be to rewrite as

Step #2
Next you find f(3) by plugging in 3 for every x.

Step #3

Now you use f(3)=10 to find g(f(3)).  Substitute 10 for f(3) and rewrite the equation as g(10) and solve.

Answer:  is not the same as 

Substitution

You can use  when you are solving problems using substitution.

For example, you can rewrite and solve the equationas where 

Other types of problems

You could also be given  and have to find what f(x) and g(x) are.

For example, find f(x) and g(x) if...

The first step would be to factor the denominator:

The next step is to find the inner function, which is this problem is g(x).  There can be multiple answers to these types of problems, but in this case, the most obvious is:

After finding g(x), you can substitute each x-5 with x.  This would leave you with the equation:

Visit the following websites for more information and examples:

http://www.purplemath.com/modules/fcncomp3.htm

http://www.mesacc.edu/~scotz47781/mat120/notes/composition/composite_functions_intro.pdf

https://www.youtube.com/watch?v=X0NzjXa2SRQ