A function is a rule that takes an input and gives exactly one output. You can imagine a function as a machine: a number goes in, the machine follows a rule, and a result comes out. Functions matter because they help describe patterns, relationships, and changes in math and science.
They are used in tables, graphs, equations, and real-world models like cost, distance, and temperature.
Key Facts
- A function pairs each input with exactly one output.
- Function notation f(x) means the output of function f when the input is x.
- If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.
- Input values are often called x-values, and output values are often called y-values.
- A relation is a function only if no input has more than one output.
- For a linear function y = mx + b, m is the rate of change and b is the starting value.
Vocabulary
- Function
- A function is a rule that assigns each input exactly one output.
- Input
- An input is the value put into a function, often represented by x.
- Output
- An output is the value produced by a function, often represented by y or f(x).
- Function notation
- Function notation, such as f(x), names a function and shows which input value is being used.
- Rate of change
- Rate of change describes how much the output changes when the input increases by 1.
Common Mistakes to Avoid
- Treating f(x) as f times x is wrong because f(x) means the output of a function named f for the input x.
- Allowing one input to have two different outputs is wrong because a function must give exactly one output for each input.
- Forgetting to follow the order of operations is wrong because a function rule like 3x + 2 must multiply before adding.
- Confusing input and output is wrong because the input goes into the rule and the output is the result after the rule is applied.
Practice Questions
- 1 A function machine uses the rule f(x) = 4x - 1. Find f(2), f(5), and f(10).
- 2 Complete the table for the rule y = 3x + 2 when x = 0, 1, 2, 3, and 4.
- 3 A relation includes the pairs (1, 4), (2, 5), (3, 6), and (2, 8). Explain whether this relation is a function and why.