Question

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

Answer 1

Domain:
`[-1,\infty)`

Range:
`(-\infty, 3]`

Decreases over:
`(-1,-\infty)`

Explanation:

Given

`f(x) = -√(x + 1) + 3` --- Missing from the question

Solving (a): The domain

To get the domain, the expression under the square root must not be negative.

In other words:

`x + 1 \ge 0`

Solve for x

`x + 1 -1\ge 0-1`

`x \ge -1`

Hence, the domain is:

`[-1,\infty)`

To get the range, we plot the values of the domain in the expression.

`x = -1`

`f(x) = -√(x + 1) + 3`

`f(-1) = -√(-1+1) +3`

`f(-1) = -√(0) +3`

`f(-1) = -0 +3`

`f(-1) = 3`

`x = \infty`

`f(\infty) = -√(\infty +1)+3`

`f(\infty) = -√(\infty)+3`

`f(\infty) = -\infty+3`

`f(\infty) = -\infty`

So, the range is:
`(-\infty, 3]`

To get the interval where the function increases or not, we simply plot the graph of f(x).

See attachment for graph.

From the attachment, it will be observed that the graph of f(x) continuously decreases from x = -1, and it never increased.

This implies that, the graph decreases over the interval
`(-1,-\infty)`