A function is a rule that takes one input and gives back exactly one output. We usually write the rule with function notation, like f(x) = 2x + 1 Here x is the input and f(x) (read "f of x") is the output. To evaluate, replace every x with the number you're given.

Domain and range

The domain is the set of all allowed inputs (the x values). The range is the set of all outputs (the f(x) values). Most functions accept every real number, but two situations shrink the domain:

• You cannot divide by zero — so any x that makes a denominator 0 is excluded.
• You cannot take a square root of a negative number — so keep whatever is under a √ at ≥ 0.

Finding a restricted domain

Look at the expression, set up the safety condition, and solve for x. For √(x − 4), you need x − 4 ≥ 0, which gives x ≥ 4. For 1/(x − 5), you need x − 5 ≠ 0, so x ≠ 5. Everything else stays in the domain. Range often comes from reading the graph or the lowest/highest output the rule can produce.