Algebra Examples

$f (x) = x^{2} - 1$ , $g (x) = 2 x$

Distribute the value across the operation.

$f (x) \div g (x)$

Evaluate $f (x)$ .

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

$f (x) = {(x)}^{2} - 1$

Simplify the result.

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Remove parentheses.

$f (x) = x^{2} - 1$

The final answer is $x^{2} - 1$ .

$x^{2} - 1$

Evaluate $g (x)$ .

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

$g (x) = 2 (x)$

Simplify the result.

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Multiply $2$ by $x$ .

$g (x) = 2 x$

The final answer is $2 x$ .

$2 x$

Compose the result function for $f (x) \div g (x)$ by replacing the function designators with the actual functions.

$(x^{2} - 1) \div 2 x$

Simplify.

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Rewrite the division as a fraction.

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

Simplify the numerator.

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Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{(x + 1) (x - 1)}{2 x}$

Set the denominator in $\frac{(x + 1) (x - 1)}{2 x}$ equal to $0$ to find where the expression is undefined.

$2 x = 0$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Divide $0$ by $2$ .

$x = 0$

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

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