The LessonA composite function is a function of a function. It combines two or more functions so that the output of one function becomes the input of another. Evaluating a composite function means putting an input into a composite function, and finding the output it relates to.
Understanding Evaluating a Composite FunctionTo evaluate a composite function means to see what output an input is mapped to. The image below shows a mapping diagram of a composite function which relates a set of inputs to a set of outputs. If we wanted to evaluate the function when the input is 2, we would see which output it is mapped to.
How to Evaluate a Composite FunctionEvaluating a composite function is easy.
QuestionTwo functions are f(x) = 2x + 1 and g(x) = x + 2. Evaluate the composite function fg(x) at x = 2.
There are two methods for evaluating the composite function.
Method 1This method is simpler.
Understand the composite function. In our example, the composite function is fg(x). Reading right to left, this means:
- the input x is passed into the function g
- which is passed into the function f
Pass the input 2 into the function g. This is evaluating the function g(x) at x = 2. Substitute x = 2 into g(x).
g(x) = x + 2
g(2) = 2 + 2
g(2) = 4
Pass the g(2) into the function f. We have found in Step 2 that g(2) = 4. Evaluate the function f(x) at x = 4 by substituting x = 4 into f(x).
f(x) = 2x + 1
f(4) = 2 × 4 + 1
f(4) = 8 + 1
f(4) = 9
Answer:The composite function fg(x) evaluated at x = 2 is: fg(2) = 9.
Method 2This method is more complicated because you find the composite function before evaluating it. The advantage is that you can then evaluate the composite function at many different values.
Find the composite function fg(x).
The composite function fg(x) = 2x + 5.
|fg(x)||Find the left most letter. It is f|
|f(x) = 2x + 1||Write out the function f(x)|
|fg(x) = 2g(x) + 1||Insert a g to the right of the f in the function name and replace x with g(x)|
|fg(x) = 2(x + 2) + 1||Substitute g(x) = x + 2 into the function (put it in brackets)|
|fg(x) = 2x + 4 + 1||Expand the brackets|
|fg(x) = 2x + 4 + 1||Collect the constant terms|
|fg(x) = 2x + 5|
Evaluate the composite function fg(x) at x = 2 by substituting x = 2 into fg(x).
fg(x) = 2x + 5
fg(2) = 2 × 2 + 5
fg(2) = 4 + 5
fg(2) = 9