# How does the range of a function relate to its graph?

Sep 10, 2014

The range of a function is its y-values or outputs. If you look at the graph from lowest point to highest point, that will be the range.

Ex: $y = {x}^{2}$ has a range of y$\ge$ 0 since the vertex is the lowest point, and it lies at (0,0). Ex: y = 2x + 1 has a range from $- \setminus \infty$ to $\setminus \infty$ since the ends of the graph point in those directions. (down and left, and up and right)
In interval notation, you would write $\left(- \setminus \infty , \setminus \infty\right)$. Ex: Some functions have interesting ranges like the sine function.
y = sin(x) Its highest values are 1 and its lowest values are -1. That range is $- 1 \le y \le 1$ or [-1,1] in interval notation.

Ex: A rather complicated function with a very challenging range is the inverse or reciprocal function, $y = \frac{1}{x}$. The output values might be difficult to describe except to say that they seem to include all real numbers except 0. (there is a horizontal asymptote on the x-axis)

You could write $\left(- \setminus \infty , 0\right) U \left(0 , \setminus \infty\right)$ in interval notation.