Question

In the following exercises, ⓐ find the domain of the function ⓑgraph the function ⓒuse the graph to determine the range.

$f\left(x\right)=\sqrt{x+1}$.

Short answer

ⓐThe domain of the give function is $[-1,\infty )$.

ⓑ Graph of given function is as following :-

ⓒ Range of the given function is$[0,\infty )$.

Step-by-step solution

Step 1. Given Information

We have given the following function:-

$f\left(x\right)=\sqrt{x+1}$

We have to find its domain. Also, we have drawn the graph of this function and the range of the function.

Step 2. Part ⓐ (Find domain of given function)

The given function is:-

$f\left(x\right)=\sqrt{x+1}$.

Here radical of this function is 2 which is even.

We know that if radical is even, then radicand is greater than or equal to zero.

So that we have:-

$$\begin{gathered} x+1\ge 0 \\ \Rightarrow x\ge -1 \end{gathered}$$

So that domain of the given function is:-

$[-1,\infty )$.

Step 3. Part ⓑ (Graph the given function)

To graph the given function$f\left(x\right)=\sqrt{x+1}$choose some points from domain $[-1,\infty )$as following :-

By using these points we can draw the graph of given function as following :-

Step 4. Range of the given function

We can see that the graph of given function lies in first and third interval.

That is y has only positive values.

So range of the given function is$[0,\infty )$.