How do you find the domain and range of #y = x^2 - 4#?

1 Answer
Bruno Pedron
Aug 5, 2015

Domain = #RR#
Range = #[-4, +oo[#

Explanation:

This quadratic function does not have a domain restriction, therefore all values of x are possible. Domain can be written as #RR# or #]-oo, +oo[#.

In order to find the range, you must find the y coordinate of the vertex. There are many ways to do that, but I prefer to start by finding the x value of the vertex before finding y.
#Xv=-b/(2a)#
Since this function doesn't have a #b# term, the vertex is exactly on the y axis (#Xv=0#). Therefore, we can substitute Xv in the law of the function and find the y value that corresponds to it:
#f(0)=0^2-4=-4#

Every quadratic function has the shape of a parabola. Since #a# is positive, the parabola has its concave up. Therefore, it's not possible to have a y value below the vertex of the function.
The range of the function is #[-4, +oo[#

Check out the graph of the function. If you have a doubt, you can always try sketching the graph.
graph{x^2-4 [-10, 10, -5, 5]}

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