Graph f(x)=1/2 square root of x-1-1

$f (x) = \frac{1}{2} \sqrt{x - 1} - 1$

Step 1

Find the domain for $y = \frac{1}{2} \cdot \sqrt{x - 1} - 1$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Set the radicand in $\sqrt{x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$x \geq 1$

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${x | x \geq 1}$

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${x | x \geq 1}$

Step 2

To find the radical expression end point, substitute the $x$ value $1$ , which is the least value in the domain, into $f (x) = \frac{\sqrt{x - 1} - 2}{2}$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{\sqrt{(1) - 1} - 2}{2}$

Simplify the result.

Simplify the numerator.

Subtract $1$ from $1$ .

$f (1) = \frac{\sqrt{0} - 2}{2}$

Rewrite $0$ as $0^{2}$ .

$f (1) = \frac{\sqrt{0^{2}} - 2}{2}$

Pull terms out from under the radical, assuming positive real numbers.

$f (1) = \frac{0 - 2}{2}$

Subtract $2$ from $0$ .

$f (1) = \frac{- 2}{2}$

Divide $- 2$ by $2$ .

$f (1) = - 1$

The final answer is $- 1$ .

$- 1$

Step 3

The radical expression end point is $(1, - 1)$ .

$(1, - 1)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

Substitute the $x$ value $2$ into $f (x) = \frac{\sqrt{x - 1} - 2}{2}$ . In this case, the point is $(2, - \frac{1}{2})$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{\sqrt{(2) - 1} - 2}{2}$

Simplify the result.

Simplify the numerator.

Subtract $1$ from $2$ .

$f (2) = \frac{\sqrt{1} - 2}{2}$

Any root of $1$ is $1$ .

$f (2) = \frac{1 - 2}{2}$

Subtract $2$ from $1$ .

$f (2) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$f (2) = - \frac{1}{2}$

The final answer is $- \frac{1}{2}$ .

$y = - \frac{1}{2}$

Substitute the $x$ value $3$ into $f (x) = \frac{\sqrt{x - 1} - 2}{2}$ . In this case, the point is $(3, \frac{\sqrt{2} - 2}{2})$ .

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{\sqrt{(3) - 1} - 2}{2}$

Simplify the result.

Subtract $1$ from $3$ .

$f (3) = \frac{\sqrt{2} - 2}{2}$

The final answer is $\frac{\sqrt{2} - 2}{2}$ .

$y = \frac{\sqrt{2} - 2}{2}$

The square root can be graphed using the points around the vertex $(1, - 1), (2, - 0.5), (3, - 0.29)$

$\begin{array}{c} x & y \\ 1 & - 1 \\ 2 & - 0.5 \\ 3 & - 0.29 \end{array}$

Step 5

Graph f(x)=1/2 square root of x-1-1

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