Question

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

Answer 1

`" "`
`color(blue)("Domain: " x>=1`, Interval Notation: `color(brown)([1, oo)`

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

Explanation:

`" "`
`color(green)"Step 1:"`

**Domain: **

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

Point to note:

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

Solve for `(x-1)>=0` to obtain `x>=1`.

Hence,

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

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

`color(green)"Step 2:"`

**Range: **

Range is the set of values of the dependent variable used in the function `f(x)` for which `f(x)` is defined.

Hence,

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

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

`color(green)"Step 3:"`

Additional note:

The function `y = f(x)= sqrt(x-1)` has no asymptotes.

Create a data table using values for `x` and corresponding values for `y`:

Observe that `Zero` and `"Negative values"` of `x` make the function `f(x)` `"undefined"` at those points.

Graph `f(x) = sqrt(x-1` to verify the results obtained: