Algebra Examples

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

Step 1

Set the denominator in $\frac{x - 2}{2 - x}$ equal to $0$ to find where the expression is undefined.

$2 - x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 3

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

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

Step 4

$y = - 1$ is a straight line perpendicular to the $y$ -axis, which means that the range is a set of one value ${y | y = - 1}$ .

Set-Builder Notation:

${y | y = - 1}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 2) \cup (2, \infty), {x | x \neq 2}$

Range: ${y | y = - 1}$

Step 6