Question

How do you find the domain and range of `-x^2+7`?

Answer 1

Domain: `RR`

Range: `(-oo,7]`

Explanation:

The domain is the set of all possible `x`-values of that function. It is the “input” values that spew out the “output” or `y`- values of the function which make up the range. The domain of a quadratic function is usually all the real numbers (`RR`). The range will vary as in this function but we will look at that below:

Let's start by taking a look at the graph `y=x^2` which, you know, is a parabola.

The domain of `y=x^2` is `RR` or `(-oo,oo)` while the range is `[0,oo)`

graph{x^2 [-10.54, 9.46, -0.56, 9.44]}

Domain: `RR`
Range: `[0,oo)`

Now let's look at the graph of `y=-x^2`. The domain remains the same but the range is now `(-oo,0]`.

graph{-x^2 [-10.17, 9.83, -9.48, 0.52]}

Domain: `RR`
Range: `(-oo,0]`

Lastly, let's look at the graph for the function given: `y=-x^2+7` which is the function `y=x^2` reflected over the `x`-axis and shifted `7` units upward:

graph{-x^2+7 [-9.88, 10.12, -2.72, 7.28]}

Domain: `RR`
Range: `(-oo,7]`

The domain is still unchanged but the range is `(-oo,7]`