Composite functions are operations that take two or more functions as one function such as h(x) = g(x). This is mainly to do with taking numbers from one set to another set.
So, for example, if a function took a number from set A to B and another function took a number from set B to C, the composite would take a number from set A directly to C.
Here is a diagram showing how the functions f(x) and g(x) can transform an input to an output.
An image which illustrates composite functions
Values in sets can be mapped to other sets by composite functions.
Properties of composite functions
There are important properties of composite functions that we should remember:
| Property name | Definition |
| Associative | If functions are composable, they are always associative. This means that no matter where the brackets are located in a function, it makes no difference to the overall function result. Therefore if f, g, h are composable then \(f(gh(x)) = (fg)h(x)\). |
| Commutative | If functions are composable it means they are not necessarily commutative. Commutativity is when swapping the order of the function's composition, does not affect it, e.g. (ab=ba). |
| One-to-one | A one-to-one composite function is where there is a single output for every input. There is also a many-to-one function where many inputs can render the same output. In the definition of a function, no composite function can be one-to-many. |
| Inverse | An inverse must exist for a composite function, therefore, there cannot be an output for which there does not exist an input. |
How do we find a composite function?
Essentially, we are carrying out a function of a function. Let's say we're trying to find \(h(x) = fg(x)\).
We would first take g(x) (the output of x) and then use it as an input in f(x) hence getting fg(x). Let's look at a worked example.
Examples of composite functions
\(f(x) = 3x + 2\) and \(g(x) = 5x -1\). If \(h(x) = fg(x)\), find the value of h(2).
Steps | Example |
Step 1: Rewrite h (x). | \(h(2) = fg(2)\) |
Step 2: Find the output of the inside function first. | \(g(2) = 5(2) - 1 = 9\) |
Step 3: Substitute this newly found output as the input in the outside function. | \(f(9) = 3(9) + 2 = 29\) |
| FINAL ANSWER | \(h(2) = 29\) |
\(f(x) = 3x + 2\) and \(g(x) = 5x -1\). If \(h(x) = fg(x)\),
find h(x). Steps | Example |
Step 1: Rewrite h (x). | \(h(x) = fg(x)\) |
Step 2: Find the output of the inside function first. | \(g(x) = 5x -1\) |
Step 3: Substitute this newly found output as the input in the outside function. | \(f(5x-1) = 3(5x-1) +2 = 15x - 3 + 2 = 15x -1\) |
| FINAL ANSWER | \(h(x) = 15x -1\) |
\(f(x) = 3x +2\). Find \(f^2(x)\).
Steps | Example |
Step 1: rewrite \(f^2(x)\). | \(f^2(x) = ff(x)\) |
Step 2: Find the output of the inside function first. | \(f(x) = 3x +2\) |
Step 3: Substitute this newly found output as the input in the outside function. | \(f(3x +2) = 3(3x+2) + 2 = 9x + 6 + 2 = 9x+8\) |
| FINAL ANSWER | \(f^2(x) = 9x+8\) |
What are some harder composite functions?
Sometimes quadratic, trigonometric and reciprocal functions can come into play, however, the logic is exactly the same as with the easier linear examples we have seen above. Let's look at some more worked examples.
\(f(x) = \cos(x), \space g(x) = 3x -2. \space h(x) = gf(x)\). Find the value of h(90).
steps | Example |
Step 1: Rewrite h (x). | \(h(90) = gf(90)\) |
Step 2: Find the output of the inside function first. | \(f(90) = \cos(90) = 0\) |
Step 3: Substitute this newly found output as the input in the outside function. | \(g(0) = 3(0) - 2 = 0-2 = -2\) |
| FINAL ANSWER | \(g(0) = -2\) |
\(f(x) = \tan^{-1}(x)\), with \(0 \leq tan^{-1}(x) \leq 2 \pi\ ), \(g(x) = x^2 + 6x -8\). \(h(x) = gf(x)\). Find the value of h(1).
steps | Example |
Step 1: Rewrite h (x). | \(h(1) = gf(1)\) |
Step 2: Find the output of the inside function first. | \(\tan^{-1}(1) = \frac{\pi}{4}\) |
Step 3: Substitute this newly found output as the input in the outside function. | \(\big(\frac{\pi}{4} \big)^2 + 6 \big( \frac{\pi}{4} \big) - 8 = -2.67076074455\) |
| FINAL ANSWER | \(h(1) = -2.67076074455\) |
\(f(x) = \tan^{-1}(x), \space g(x) = 3-x^2. \space h(x) = gf^{-1}(x)\). Find \(h(x)\)
steps | Example |
Step 1: Rewrite h (x). | \(h(x) = gf^{-1}(x)\) |
Step 2: Find the output of the inside function first. | \(f^{-1}(x) = \tan(x)\) |
Step 3: Substitute this newly found output as the input in the outside function. | \(g(\tan(x)) = 2 -\tan^2(x)\) |
| FINAL ANSWER | \(h(x) = 3 - \tan^2(x)\) |
Composite Functions - Key takeaways
- Composite functions are when a combination of functions are undertaken to give a common output.
- Always perform the inside function first before inputting this value into the outside function.
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