Calculus Examples

$f (x) = \frac{x}{2 x - \frac{2}{3}}$ , $(- 14, 0)$

Step 1

Evaluate $f (x) = \frac{x}{2 x - \frac{2}{3}}$ at $x = - 14$ .

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Step 1.1

Replace the variable $x$ with $- 14$ in the expression.

$f (- 14) = \frac{- 14}{2 (- 14) - \frac{2}{3}}$

Step 1.2

Simplify the result.

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Step 1.2.1

Cancel the common factor of $- 14$ and $2 (- 14) - \frac{2}{3}$ .

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Step 1.2.1.1

Factor $2$ out of $- 14$ .

$f (- 14) = \frac{2 (- 7)}{2 (- 14) - \frac{2}{3}}$

Step 1.2.1.2

Cancel the common factors.

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Step 1.2.1.2.1

Factor $2$ out of $- \frac{2}{3}$ .

$f (- 14) = \frac{2 (- 7)}{2 (- 14) + 2 (- \frac{1}{3})}$

Step 1.2.1.2.2

Factor $2$ out of $2 (- 14) + 2 (- \frac{1}{3})$ .

$f (- 14) = \frac{2 (- 7)}{2 (- 14 - \frac{1}{3})}$

Step 1.2.1.2.3

Cancel the common factor.

$f (- 14) = \frac{2 \cdot - 7}{2 (- 14 - \frac{1}{3})}$

Step 1.2.1.2.4

Rewrite the expression.

$f (- 14) = \frac{- 7}{- 14 - \frac{1}{3}}$

Step 1.2.2

Simplify the denominator.

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Step 1.2.2.1

To write $- 14$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 14) = \frac{- 7}{- 14 \cdot \frac{3}{3} - \frac{1}{3}}$

Step 1.2.2.2

Combine $- 14$ and $\frac{3}{3}$ .

$f (- 14) = \frac{- 7}{\frac{- 14 \cdot 3}{3} - \frac{1}{3}}$

Step 1.2.2.3

Combine the numerators over the common denominator.

$f (- 14) = \frac{- 7}{\frac{- 14 \cdot 3 - 1}{3}}$

Step 1.2.2.4

Simplify the numerator.

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Step 1.2.2.4.1

Multiply $- 14$ by $3$ .

$f (- 14) = \frac{- 7}{\frac{- 42 - 1}{3}}$

Step 1.2.2.4.2

Subtract $1$ from $- 42$ .

$f (- 14) = \frac{- 7}{\frac{- 43}{3}}$

Step 1.2.2.5

Move the negative in front of the fraction.

$f (- 14) = \frac{- 7}{- \frac{43}{3}}$

Step 1.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (- 14) = - 7 (- \frac{3}{43})$

Step 1.2.4

Multiply $- 7 (- \frac{3}{43})$ .

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Step 1.2.4.1

Multiply $- 1$ by $- 7$ .

$f (- 14) = 7 (\frac{3}{43})$

Step 1.2.4.2

Combine $7$ and $\frac{3}{43}$ .

$f (- 14) = \frac{7 \cdot 3}{43}$

Step 1.2.4.3

Multiply $7$ by $3$ .

$f (- 14) = \frac{21}{43}$

Step 1.2.5

The final answer is $\frac{21}{43}$ .

$\frac{21}{43}$

Step 2

Since $\frac{21}{43}$ $\neq$ $0$ , the point is not on the graph.

The point is not on the graph

Step 3