Trigonometry Examples

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

Step 1

Find the domain for $y = 3 - \sqrt{x - 2}$ 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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Step 1.1

Set the radicand in $\sqrt{x - 2}$ greater than or equal to $0$ to find where the expression is defined.

$x - 2 \geq 0$

Step 1.2

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 1.3

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

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $2$ from $2$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

$f (2) = 3 - 0$

Step 2.2.1.4

Multiply $- 1$ by $0$ .

$f (2) = 3 + 0$

Step 2.2.2

Add $3$ and $0$ .

$f (2) = 3$

Step 2.2.3

The final answer is $3$ .

$3$

Step 3

The radical expression end point is $(2, 3)$ .

$(2, 3)$

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.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Subtract $2$ from $3$ .

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

Step 4.1.2.1.2

Any root of $1$ is $1$ .

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

Step 4.1.2.1.3

Multiply $- 1$ by $1$ .

$f (3) = 3 - 1$

Step 4.1.2.2

Subtract $1$ from $3$ .

$f (3) = 2$

Step 4.1.2.3

The final answer is $2$ .

$y = 2$

Step 4.2

Substitute the $x$ value $4$ into $f (x) = 3 - \sqrt{x - 2}$ . In this case, the point is $(4, 3 - \sqrt{2})$ .

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Step 4.2.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = 3 - \sqrt{(4) - 2}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Subtract $2$ from $4$ .

$f (4) = 3 - \sqrt{2}$

Step 4.2.2.2

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

$y = 3 - \sqrt{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(2, 3), (3, 2), (4, 1.59)$

$\begin{array}{c} x & y \\ 2 & 3 \\ 3 & 2 \\ 4 & 1.59 \end{array}$

Step 5