Question

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

Answer 1

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)`