What are Composite Functions? Definition and Examples for GCSE Maths

📚 Quick Study Guide

🔢 Definition: A composite function is a function that is applied to the result of another function. It's like a function within a function! For example, if you have two functions, $f(x)$ and $g(x)$, the composite function $f(g(x))$ means you first apply $g$ to $x$, and then apply $f$ to the result.

✍️ Notation: Composite functions are written as $f(g(x))$, which is read as "f of g of x". Another notation for $f(g(x))$ is $(f \circ g)(x)$.

⚙️ Order Matters: The order in which you apply the functions is crucial. $f(g(x))$ is generally not the same as $g(f(x))$.

💡 How to Solve: To evaluate a composite function, start with the innermost function and work your way outwards. For example, to find $f(g(2))$, first find $g(2)$, and then substitute that result into $f(x)$.

Practice Quiz

1. What is the value of $f(g(x))$ if $f(x) = x + 2$ and $g(x) = 3x$?
1. $3x + 2$
2. $3x + 6$
3. $x + 5$
4. $6x + 2$
2. Given $f(x) = x^2$ and $g(x) = x - 1$, find $f(g(2))$.
1. 1
2. 3
3. 4
4. 9
3. If $f(x) = 2x - 3$ and $g(x) = x + 5$, what is $g(f(x))$?
1. $2x + 2$
2. $2x - 8$
3. $2x + 7$
4. $2x - 2$
4. Let $f(x) = \frac{1}{x}$ and $g(x) = x + 2$. Find $f(g(x))$.
1. $\frac{1}{x+2}$
2. $\frac{1}{x} + 2$
3. $\frac{x}{1+2x}$
4. $x + \frac{1}{2}$
5. Given $f(x) = \sqrt{x}$ and $g(x) = 4x$, what is $f(g(9))$?
1. 6
2. 12
3. 36
4. 2
6. If $f(x) = 5x$ and $g(x) = x^2 - 1$, find $g(f(x))$.
1. $5x^2 - 1$
2. $25x^2 - 1$
3. $5(x^2 - 1)$
4. $x^2 - 5$
7. What is $f(f(x))$ if $f(x) = x - 3$?
1. $x - 6$
2. $x - 9$
3. $x^2 - 6x + 9$
4. $2x - 6$