How do you find the domain and range of #f(x) = x^2 + 5#?

1 Answer
Jun 9, 2016

Answer:

Domain: #x in (-oo, oo)#
Range: #f(x) in (5, oo)#

Explanation:

The domain is easy. You can enter any real number without restriction i.e. #x in RR# or equivalently #x in (-oo, oo)#.

The range is a little trickier, but you can arrive at the answer two ways. First, if you know the graph of the function, you would know that it is a parabola opening up with its vertex at (0,5). Hence #f(x) in (5, oo)#.

graph{x^2+5 [-9.66, 10.34, -0.96, 9.04]}

Another way to think about the range is to consider what values #x# can be. Because the variable is squared and a positive number is being added, the output must always be positive. Additionally, the smallest value will be achieved when #x=0#. At this point, #f(x)=5# and so #f(x) in (5, oo)#