EDEXCEL GCSE HIGHER MATHS - What are Compound and Composite Functions

Edexcel GCSE Higher Maths: Compound and Composite Functions

What are Compound Functions?

A compound function is simply a function that is made up of two or more simpler functions. These functions are combined by applying one function to the result of another.

Example:

Imagine you have two functions:

• f(x) = x + 2 (This adds 2 to the input)
• g(x) = 2x (This doubles the input)

We can combine these to form a compound function. Let's say we want to apply g(x) to the result of f(x). This would be written as:

g(f(x))

To find the value of this compound function, we first apply f(x) to the input, then apply g(x) to the result.

For example, if x = 3:

1. f(3) = 3 + 2 = 5
2. g(5) = 2 * 5 = 10

Therefore, g(f(3)) = 10

What are Composite Functions?

A composite function is a type of compound function where the output of one function becomes the input of another.

Example:

Using the same functions as before:

• f(x) = x + 2
• g(x) = 2x

The composite function f(g(x)) would be formed by first applying g(x) to the input, and then applying f(x) to the result.

For example, if x = 3:

1. g(3) = 2 * 3 = 6
2. f(6) = 6 + 2 = 8

Therefore, f(g(3)) = 8

Key Differences:

• Compound Functions: Functions can be applied in any order, not necessarily one after the other.
• Composite Functions: The output of the first function is always the input of the second function.

Notation and Terminology

• f(g(x)) is read as "f of g of x"
• The function that is applied first is called the inner function.
• The function that is applied second is called the outer function.

Finding the Rule for a Composite Function

To find the rule for a composite function, we need to substitute the entire inner function into the outer function.

Example:

Find the rule for the composite function f(g(x)) given:

• f(x) = 3x - 1

• g(x) = x^2 + 2

• Substitute g(x) into f(x): f(g(x)) = 3(x^2 + 2) - 1

• Simplify the expression: f(g(x)) = 3x^2 + 6 - 1 f(g(x)) = 3x^2 + 5

Therefore, the rule for the composite function f(g(x)) is 3x^2 + 5.

Practice Questions

1. If f(x) = x - 3 and g(x) = 2x + 1, find:

   • f(g(x))
   • g(f(x))

2. Find the rule for the composite function h(k(x)), given:

   • h(x) = x^3
   • k(x) = 2x - 1

3. If f(g(2)) = 7 and g(2) = 4, find f(4).

Remember: Practice is key to mastering compound and composite functions. Solve as many practice problems as you can to solidify your understanding!