Functions – Rules That Connect Inputs to Outputs
A function is a rule that assigns exactly one output to every input. Functions are the heart of mathematics — they describe how one quantity depends on another, from temperature-time relationships to population growth.
What Is a Function?
A function takes an input value, applies a rule, and produces a single output. Every input gives exactly one output. Functions are often written as f(x), read as 'f of x', where x is the input.
Notation and Language
| Notation | Meaning |
|---|---|
| f(x) = 2x + 1 | The rule is: double the input and add 1 |
| f(3) | The output when x = 3 |
| Domain | The set of all valid inputs |
| Range | The set of all possible outputs |
Evaluating Functions
f(4) = 3(4) - 2 = 12 - 2 = 10.
g(-3) = (-3) squared + 1 = 9 + 1 = 10.
h(0) = 0 - 0 + 3 = 3. h(2) = 8 - 10 + 3 = 1.
Composite Functions
A composite function applies one function and then another. Written as fg(x) or f(g(x)), it means: first apply g to x, then apply f to the result.
fg(x) = f(g(x)) = f(3x) = 3x + 2. gf(x) = g(f(x)) = g(x+2) = 3(x+2) = 3x + 6. Note: fg and gf are not the same!
Inverse Functions
The inverse function undoes what the original function did. If f(x) = y then f inverse(y) = x.
Let y = 2x - 5. Rearrange for x: 2x = y + 5, so x = (y+5)/2. The inverse is f inverse(x) = (x + 5) / 2.
Real-Life Application
A taxi costs f(x) = 2x + 3 where x is the distance in kilometres. The inverse function finds how far you can travel for a given cost.
Key Takeaways
- A function gives exactly one output for each input: f(x) maps x to a unique value.
- Evaluate by substituting the input for x and calculating.
- Composite functions apply two rules in sequence; order matters.
- Inverse functions reverse the mapping: they undo the original function.
Practice Questions
- f(x) = 5x - 3. Find f(2) and f(-1).
- g(x) = x squared - 2x + 4. Find g(3).
- f(x) = x + 4 and g(x) = 2x. Find fg(5) and gf(5).
- Find the inverse of f(x) = 3x + 6.
- A function gives the cost in pounds of renting a hall: C(h) = 50h + 80 where h is the number of hours. Find C(3) and find h when C(h) = 280.