A function is one of the most powerful ideas in mathematics — and it's actually very intuitive once you see what it's really saying.
You should be comfortable with basic algebra — substituting values into expressions, and the idea of a variable. That's all you need.
A function is a rule that assigns exactly one output to each input. Think of it like a machine: you drop in a number, and you always get exactly one number out — never two, never zero.
The key word is exactly one. If an input could give two different outputs, the rule isn't a function.
We write functions using function notation. Instead of writing y = 2x + 3, we write:
Read this as "f of x equals 2x + 3." The letter f is the name of the function. The x is the input. We can use any letter — g(x), h(t), A(r) are all function names.
A relation is any pairing of inputs and outputs. A function is a special relation where each input has exactly one output.
Each input has exactly one output.
Input 1 has TWO outputs (4 and 7).
If you have a graph, there's a quick way to check if it's a function:
Draw (or imagine) any vertical line across the graph. If the vertical line ever touches the graph in more than one point, the graph does NOT represent a function.
The parabola on the left passes the vertical line test — it's a function. The circle on the right fails — it is not a function.
Given the set of pairs: {(1, 5), (2, 10), (3, 15), (4, 5)}
Note: It's fine if two different inputs map to the same output (1 → 5 and 4 → 5). That's still a function! The restriction only applies to inputs — each input can only go to one output.
Q1. Is {(2, 6), (3, 6), (4, 8)} a function? Why or why not?
Yes, it is a function. Each input (2, 3, 4) appears exactly once. The fact that inputs 2 and 3 both output 6 is fine — inputs can share an output.
Q2. Is {(1, 2), (1, 3), (2, 4)} a function? Why or why not?
No, it is NOT a function. The input 1 is paired with two different outputs (2 and 3). This violates the rule that each input must have exactly one output.