# What is the domain and range for f(x)=sqrt(x-1)?

May 10, 2018

$\text{ }$
color(blue)("Domain: " x>=1, Interval Notation: color(brown)([1, oo)

color(blue)("Range: " f(x)>=0, Interval Notation: color(brown)([0, oo)

#### Explanation:

$\text{ }$
$\textcolor{g r e e n}{\text{Step 1:}}$

Domain:

The domain of the given function $f \left(x\right)$ is the set of input values for which $f \left(x\right)$ is real and defined.

Point to note:

color(red)(sqrt(f(x)) = f(x)>=0

Solve for $\left(x - 1\right) \ge 0$ to obtain $x \ge 1$.

Hence,

color(blue)("Domain: " x>=1

Interval Notation: color(brown)([1, oo)

$\textcolor{g r e e n}{\text{Step 2:}}$

Range:

Range is the set of values of the dependent variable used in the function $f \left(x\right)$ for which $f \left(x\right)$ is defined.

Hence,

color(blue)("Range: " f(x)>=0

Interval Notation: color(brown)([0, oo)

$\textcolor{g r e e n}{\text{Step 3:}}$

The function $y = f \left(x\right) = \sqrt{x - 1}$ has no asymptotes.
Create a data table using values for $x$ and corresponding values for $y$:
Observe that $Z e r o$ and $\text{Negative values}$ of $x$ make the function $f \left(x\right)$ $\text{undefined}$ at those points.
Graph f(x) = sqrt(x-1 to verify the results obtained: