Question

State the domain, the range, and the intervals on which function is increasing decreasing, or constant in interval notation. State the domain and range using interval notation.

Answer 1

`Domain = (-\infty, \infty)`

`Range = (-\infty, 4]`

Increasing interval is:
`(-\infty, 0)`

Decreasing interval is:
`(0, \infty)`

Constant at no interval

Explanation:

Given

See attachment for graph

Solving (a): The domain

From, the attached graph, we have:

`y = -x^2 + 4`

The degree of the polynomial (i.e 2) is even.

Hence, the domain is the set of all real numbers, i.e.

`Domain = (-\infty, \infty)`

Solving (b): The range

The curve of
`y = -x^2 + 4` opens downward, and the maximum is:

`y_(max) = 4`

This means that the minimum is:

`y_(mi n) = -\infty`

Hence, the range of the set is:

`Range = (-\infty, 4]`

Solving (c): Interval where the function increases/decreases/constant

In (a), we have:

`Domain = (-\infty, \infty)`

Split to 2 (at vertex x = 0)

`(-\infty, 0)` and
`(0, \infty)`

So:

The increasing interval is:
`(-\infty, 0)`

The decreasing interval is:
`(0, \infty)`

The function has a tangent at
`x = 0` but at no interval, was the function constant