Functions are tested heavily on the SAT — expect 4-6 questions per test. Here is everything you need to know.
Function Notation
does NOT mean times . It means "the output of function when the input is ."
If :
Domain and Range
Domain: All possible inputs (x-values). Ask: "What values of make this function break?"
Common restrictions:
- Division by zero: — domain excludes
- Square root of negative: — domain is
Range: All possible outputs (y-values).
Function Transformations
Starting from :
| Transformation | Effect |
|---|---|
| Shift UP by | |
| Shift DOWN by | |
| Shift LEFT by | |
| Shift RIGHT by | |
| Reflect over x-axis | |
| Reflect over y-axis | |
| with | Vertical stretch |
| with | Vertical compress |
Memory trick: Changes INSIDE the parentheses affect (horizontal, opposite direction). Changes OUTSIDE affect (vertical, same direction).
Reading Function Graphs
On the SAT, you will often see a graph of and be asked to find . Just go to on the graph and read the -value.
For : find where the graph crosses the x-axis.
For : find the y-intercept.
Common Mistakes to Avoid
- shifts LEFT, not right: Inside the parentheses, everything is backwards.
- does NOT mean : Functions can have the same output for different inputs (think parabolas).
- Confusing with : asks for x-intercepts. asks for the y-intercept.
SAT Pro Tip
When the SAT gives you an unfamiliar function (like a table of values or a weird graph), do not panic. Just plug in the given input and read the output. Functions on the SAT are always about input-output relationships — nothing more.
Master functions with our Function Notation lessons with practice exercises.