Precalculus Examples

$f (x) = \frac{1}{x}$ , $\frac{f (x) - f (a)}{x - a}$

Step 1

Evaluate the numerator in $\frac{f (x) - f a}{x - a}$ .

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Step 1.1

Replace the function designators in $f (x) - f a$ with the actual functions.

$(\frac{1}{x}) - \frac{1}{x} a$

Step 1.2

Simplify $(\frac{1}{x}) - \frac{1}{x} a$ .

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Step 1.2.1

Combine $a$ and $\frac{1}{x}$ .

$\frac{1}{x} - \frac{a}{x}$

Step 1.2.2

Reorder $\frac{1}{x}$ and $- \frac{a}{x}$ .

$- \frac{a}{x} + \frac{1}{x}$

Step 2

Evaluate the denominator in $\frac{f (x) - f a}{x - a}$ .

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Step 2.1

Replace the function designators in $x - a$ with the actual functions.

$x - a$

Step 2.2

Reorder $x$ and $- a$ .

$- a + x$

Step 3

Rewrite $\frac{f (x) - f a}{x - a}$ .

$\frac{- \frac{a}{x} + \frac{1}{x}}{- a + x}$

Step 4

Simplify $\frac{- \frac{a}{x} + \frac{1}{x}}{- a + x}$ .

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Step 4.1

Multiply the numerator and denominator of the fraction by $x$ .

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Step 4.1.1

Multiply $\frac{- \frac{a}{x} + \frac{1}{x}}{- a + x}$ by $\frac{x}{x}$ .

$\frac{x}{x} \cdot \frac{- \frac{a}{x} + \frac{1}{x}}{- a + x}$

Step 4.1.2

Combine.

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

Step 4.2

Apply the distributive property.

$\frac{x (- \frac{a}{x}) + x \frac{1}{x}}{x (- a) + x \cdot x}$

Step 4.3

Simplify by cancelling.

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Step 4.3.1

Cancel the common factor of $x$ .

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Step 4.3.1.1

Move the leading negative in $- \frac{a}{x}$ into the numerator.

$\frac{x \frac{- a}{x} + x \frac{1}{x}}{x (- a) + x \cdot x}$

Step 4.3.1.2

Cancel the common factor.

$\frac{x \frac{- a}{x} + x \frac{1}{x}}{x (- a) + x \cdot x}$

Step 4.3.1.3

Rewrite the expression.

$\frac{- a + x \frac{1}{x}}{x (- a) + x \cdot x}$

Step 4.3.2

Cancel the common factor of $x$ .

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Step 4.3.2.1

Cancel the common factor.

$\frac{- a + x \frac{1}{x}}{x (- a) + x \cdot x}$

Step 4.3.2.2

Rewrite the expression.

$\frac{- a + 1}{x (- a) + x \cdot x}$

Step 4.4

Simplify terms.

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Step 4.4.1

Factor $x$ out of $x (- a) + x \cdot x$ .

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Step 4.4.1.1

Factor $x$ out of $x \cdot x$ .

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

Step 4.4.1.2

Factor $x$ out of $x (- a) + x (x)$ .

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

Step 4.4.2

Factor $- 1$ out of $- a$ .

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

Step 4.4.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.4.4

Factor $- 1$ out of $- (a) - 1 (- 1)$ .

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

Step 4.4.5

Rewrite $- (a - 1)$ as $- 1 (a - 1)$ .

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

Step 4.4.6

Factor $- 1$ out of $- a$ .

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

Step 4.4.7

Factor $- 1$ out of $x$ .

$\frac{- 1 (a - 1)}{x (- (a) - 1 (- x))}$

Step 4.4.8

Factor $- 1$ out of $- (a) - 1 (- x)$ .

$\frac{- 1 (a - 1)}{x (- (a - x))}$

Step 4.4.9

Rewrite $- (a - x)$ as $- 1 (a - x)$ .

$\frac{- 1 (a - 1)}{x (- 1 (a - x))}$

Step 4.4.10

Cancel the common factor.

$\frac{- 1 (a - 1)}{x (- 1 (a - x))}$

Step 4.4.11

Rewrite the expression.

$\frac{a - 1}{x (a - x)}$