Graph m(x)=2 square root of x-1-3

$m (x) = 2 \sqrt{x - 1} - 3$

Find the domain for $y = 2 \sqrt{x - 1} - 3$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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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:

${x | x \geq 1}$

To find the radical expression end point, substitute the $x$ value $1$ , which is the least value in the domain, into $f (x) = 2 \sqrt{x - 1} - 3$ .

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

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

Simplify the result.

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Simplify each term.

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Subtract $1$ from $1$ .

$f (1) = 2 \sqrt{0} - 3$

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

$f (1) = 2 \sqrt{0^{2}} - 3$

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

$f (1) = 2 \cdot 0 - 3$

Multiply $2$ by $0$ .

$f (1) = 0 - 3$

Subtract $3$ from $0$ .

$f (1) = - 3$

The final answer is $- 3$ .

$- 3$

The radical expression end point is $(1, - 3)$ .

$(1, - 3)$

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.

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

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

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

Simplify the result.

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Simplify each term.

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Subtract $1$ from $2$ .

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

Any root of $1$ is $1$ .

$f (2) = 2 \cdot 1 - 3$

Multiply $2$ by $1$ .

$f (2) = 2 - 3$

Subtract $3$ from $2$ .

$f (2) = - 1$

The final answer is $- 1$ .

$y = - 1$

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

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

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

Simplify the result.

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Subtract $1$ from $3$ .

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

The final answer is $2 \sqrt{2} - 3$ .

$y = 2 \sqrt{2} - 3$

The square root can be graphed using the points around the vertex $(1, - 3), (2, - 1), (3, - 0.17)$

$\begin{array}{c} x & y \\ 1 & - 3 \\ 2 & - 1 \\ 3 & - 0.17 \end{array}$

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