Question

find the domain and the range of the function of f(x)= square roots x+5.

Answer 1

To find the domain and range of the function \( f(x) = \sqrt{x + 5} \), we need to consider where the function is defined.

### Domain
The expression inside the square root, \( x + 5 \), must be non-negative for \( f(x) \) to be defined. Therefore, we need:

\[
x + 5 \geq 0
\]

Solving for \( x \):

\[
x \geq -5
\]

Thus, the domain of \( f(x) \) is:

\[
\text{Domain: } x \in [-5, \infty)
\]

### Range
Next, we determine the range of \( f(x) \). The square root function produces non-negative outputs.

1. When \( x = -5 \):
\[
f(-5) = \sqrt{-5 + 5} = \sqrt{0} = 0
\]

2. As \( x \) increases beyond \(-5\), \( f(x) \) will increase without bound.

Therefore, \( f(x) \) starts at 0 and goes to \( \infty \).

Thus, the range of \( f(x) \) is:

\[
\text{Range: } f(x) \in [0, \infty)
\]

### Summary
- **Domain:** \( [-5, \infty) \)
- **Range:** \( [0, \infty) \)