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

1 Answer
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#>=# 0 since the vertex is the lowest point, and it lies at (0,0).
my screenshot

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)#.

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Ex: Some functions have interesting ranges like the sine function.
y = sin(x) my screenshot3
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}#.

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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!