Understanding Evaluating a Composite Function
To 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 Function
Evaluating a composite function is easy.Question
Two 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 1
This method is simpler.Step-by-Step:
1
Understand the composite function.
In our example, the composite function is fg(x). Reading right to left, this means:
Because we are evaluating the function at x = 2, we will pass in 2 as an input, rather than x.
- the input x is passed into the function g
- which is passed into the function f
Because we are evaluating the function at x = 2, we will pass in 2 as an input, rather than x.
2
Pass the input 2 into the function g.
This is evaluating the function g(x) at x = 2.
Substitute x = 2 into g(x).
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
3
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).
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 2
This 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.Step-by-Step:
1
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 |
2
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
Answer:
The composite function fg(x) evaluated at x = 2 is: fg(2) = 9. Both methods give the same answer: if the functions f(x) = 2x + 1 and g(x) = x + 2 are combined into a composite function fg(x) and evaluated at 2, the answer is 9.A Note on Notation
If a composite function fg is evaluated at a number (e.g. 2), this is denoted fg(2).
This page was written by Stephen Clarke.
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