Question

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

Answer 1

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`>=` 0 since the vertex is the lowest point, and it lies at (0,0).

Ex: y = 2x + 1 has a range from `-\infty` to `\infty` since the ends of the graph point in those directions. (down and left, and up and right)
In interval notation, you would write `(-\infty,\infty)`.

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<=y<=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 `(-\infty,0)U(0,\infty)` in interval notation.

Enjoy your study of range!