# Use the Vertical Line Test to Identify a Function

A function is basically anything you can graph on your graphing calculator in “*y* =” or graphing mode. The line *y* = 3*x* – 2 is a function, as is the parabola *y* = 4*x*^{2} – 3*x* + 6. On the other hand, the sideways parabola *x* = 5*y*^{2} + 4*y* – 10 isn’t a function because there’s no way to write it as *y* = *something. *(Unless you write *y* = ± *something*, which doesn’t count.)

You can determine whether or not the graph of a curve is a function with the vertical line test. If there’s no place on the graph where you could draw a vertical line that touches the curve more than once, then it is a function. And if you can draw a vertical line anywhere on the graph that touches the curve more than once, then it is not a function.

You can rewrite the above functions using *f*(*x*) or *g*(*x*) instead of *y*. This changes nothing; using something like *f*(*x*) is just a convenient notation. Here’s a sampling of calculus functions:

Virtually every single calculus problem involves functions in one way or another. To review some function basics, try the following examples.

## Practice questions

- If
*f*(*x*) = 3*x*^{2}– 4*x*+ 8, what does*f*(*a*+*b*) equal? - For the line
*g*(*x*) =

## Answers and explanations

1. **3 a^{2} + 6ab + 3b^{2} – 4a – 4b + 8**

2. **The slope is –4 and the y intercept is 5. **