What is a function?
Some quantities depend on others. The cost of pens depends on how many you buy. The input is the variable we choose. The output is what we get.
A function is a rule that gives exactly one output for each input. If one input could give two different outputs, it is not a function.
- “Double the number” is a function: 5 → 10.
- “A number whose square is 9” is not: 9 could come from 3 or −3, so the backward rule gives two answers.
Function notation, image and preimage
We name a function with a letter, often f. We write f(x) = 2x + 1. Read it as “f of x equals two x plus one”.
- f(3) means “put 3 in”: f(3) = 2 × 3 + 1 = 7.
- 7 is the image of 3.
- 3 is a preimage of 7. To find a preimage, solve an equation: f(x) = 7.
Note: f(x) does not mean f times x. It is a name plus an input.
Domain and range
The domain is the set of inputs you are allowed to use. The range is the set of outputs you actually get. For the number of tickets sold, the domain is whole numbers only — you cannot sell 2.5 tickets.
Four ways to show a function
The same function can be shown in four ways. Learn to switch between them.
- Words: “double the number and add one”.
- Formula: f(x) = 2x + 1.
- Table: x = 0, 1, 2, 3 gives f(x) = 1, 3, 5, 7.
- Graph: plot the points (x, f(x)) and join them.
Vertical line test: on a graph, draw any up-down line. If it ever cuts the graph twice, one input has two outputs, so it is not a function.
To compare two functions, compare their tables or graphs: which one grows faster? Where are they equal?
Reading a graph: increasing, decreasing, maximum, minimum
Read a graph from left to right, like a sentence.
- Increasing: the graph goes up — bigger x gives bigger output.
- Decreasing: the graph goes down.
- Maximum: the highest point. Minimum: the lowest point.
For h(x) = 6x − x² on 0 ≤ x ≤ 6: increasing from 0 to 3, maximum 9 at x = 3, decreasing from 3 to 6. A variation table writes this as arrows: ↗ up to 9, then ↘.
A sign table shows where f(x) is positive (+), zero (0) or negative (−). For f(x) = x − 2: negative when x < 2, zero at x = 2, positive when x > 2.
Functions as models of real life
A model is a function that describes something real. A phone plan costs ₹99 plus ₹2 per GB: C(g) = 99 + 2g.
Rate of change tells how fast the output changes per unit of input. From g = 1 to g = 5, C rises from 101 to 109: rate = 8 ÷ 4 = ₹2 per GB.
Optimisation means finding the best value — often a maximum or minimum. If a ball's height is h(t) = 6t − t², the ball is highest at t = 3 s.
Try it
Walk 10 steps, then 20, then 30 and time each walk with a phone. Make a table of steps → seconds. Is it a function? Draw its graph. Then open the free-play step in the 3D and test each rule.
Key formulas and definitions
- f(x) = rule; f(a) = image of a
- Preimage of b: solve f(x) = b
- Rate of change = (change in output) ÷ (change in input)
- Point on graph: (x, f(x))
Worked examples
1. f(x) = 2x + 1. Find f(5).
Put 5 in place of x: f(5) = 2 × 5 + 1 = 10 + 1 = 11.
2. g(x) = x² − 3. Find g(−2).
g(−2) = (−2)² − 3 = 4 − 3 = 1. Remember (−2)² = +4.
3. f(x) = 3x − 4. Find the preimage of 11.
Solve 3x − 4 = 11. Add 4: 3x = 15. Divide by 3: x = 5. So 5 is the preimage of 11.
4. Make a table for f(x) = 10 − x for x = 0, 2, 4, 6 and say if it is increasing or decreasing.
f(0) = 10, f(2) = 8, f(4) = 6, f(6) = 4. As x grows, f(x) falls, so f is decreasing.
5. A taxi charges ₹50 plus ₹12 per km. Write the function and find the fare for 8 km.
F(d) = 50 + 12d. F(8) = 50 + 96 = ₹146.
6. A ball's height is h(t) = 6t − t² metres after t seconds. Find the maximum height and when it happens.
Table: h(0)=0, h(1)=5, h(2)=8, h(3)=9, h(4)=8, h(5)=5, h(6)=0. It rises to 9 m at t = 3 s, then falls. Maximum height = 9 m at t = 3 s.
Common mistakes
- Reading f(x) as “f times x”. It means “the output of f when the input is x”.
- Forgetting brackets with negatives: for g(x) = x², g(−3) = (−3)² = 9, not −9.
- Mixing up image and preimage: the image is the output; the preimage is the input.
- Thinking every graph is a function. If a vertical line cuts it twice, it is not.