Table of Contents |
You may have noticed that when we evaluate a function, we typically evaluate it for a number or variable, but we can go one step further and evaluate a function of another function. This process is very similar to when you would substitute a variable with an expression.
EXAMPLE
Substitute into and solve for x.Substitute in for y | |
Distribute subtraction | |
Combine like terms | |
Add 1 to both sides | |
Multiply both sides by -1 | |
Our Solution |
When working with composite functions, we follow a similar process. If we are given two functions and and we want to find we simply replace all instances of x in the function with whatever is set equal to. Then we simplify.
Let's take a look at how to actually evaluate a composite function.
EXAMPLE
Suppose and . Find .Replace x with | |
Replace with | |
Expand | |
FOIL | |
Evaluate | |
Our solution |
Sometimes we may be asked to solve a particular composite function when x is equal to a given value, for example . To make this evaluation, we follow the same steps we did above then at the end we simply substitute the value we are given for x. Let's see how we do this:
EXAMPLE
Using the result from above, evaluate .Replace x with 2 | |
Evaluate | |
Simplify | |
Our solution |
Evaluate | |
Simplify | |
Evaluate by plugging in 1 for | |
Evaluate | |
Simplify | |
Our Solution |
Sometimes you may come across a situation where you will need to evaluate a function for itself, namely . This simply means that you are plugging in the expression of the function for each x in the function.
EXAMPLE
Find for .Replace x with | |
Replace with on the right side | |
Distribute 2 into | |
Simplify | |
Our solution |
Like with the previous problem, you may be asked to find the composite function of itself when x is a given value, for example . Just like before we simply need to replace each instance of x in the function with the given value.
EXAMPLE
Using the result from above, evaluate.Replace x with 0 | |
Evaluate | |
Simplify | |
Our solution |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License