Precalculus Examples

$x - 2 = {(y - 2)}^{2}$

Step 1

Rewrite the equation as ${(y - 2)}^{2} = x - 2$ .

${(y - 2)}^{2} = x - 2$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.2

Add $2$ to both sides of the equation.

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

Step 3.3

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.4

Add $2$ to both sides of the equation.

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

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

Step 4

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 5

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 6

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

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Step 7

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 8

Determine the domain and range.

Domain: $[2, \infty), {x | x \geq 2}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 9