Algebra · Quick Reference

Functions Cheat Sheet

Algebra I · Lesson 4/17

In one line: a function is a rule that assigns exactly one output to every input — one of the most important ideas in all of mathematics, since it lets us describe relationships precisely.

Key Ideas

1Function Definition. A relationship where every input (x-value) produces exactly one output (y-value).

2Function Notation. f(x) is read 'f of x' and represents the output of function f for input x. f(3) means 'evaluate the function at x=3'.

3The Vertical Line Test. If any vertical line crosses a graph more than once, the graph does NOT represent a function.

4Domain and Range. The domain is the complete set of possible input (x) values; the range is the complete set of possible output (y) values.

5Evaluating Functions. Substitute the given input value everywhere the variable appears in the function's formula, then simplify.

Worked Examples

If f(x) = 2x + 3, find f(5).

f(5) = 13

If g(x) = x^2 - 4, find g(-3).

g(-3) = 5

Does the equation x = y^2 represent a function of x? Use the vertical line test reasoning.

No, x=y^2 does not represent y as a function of x

Practice Yourself

If f(x) = 3x - 1, find f(4).

If f(x) = x^2 + 2, find f(3).

If f(x) = 5x, find f(0).

What test determines if a graph represents a function?

The vertical line test