Precalculus Examples

$f (x) = \frac{1}{x - 2}$

To find the x-intercept, substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{1}{x - 2}$

Solve the equation.

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Rewrite the equation as $\frac{1}{x - 2} = 0$ .

$\frac{1}{x - 2} = 0$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 2, 1$

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $x - 2$ is $x - 2$ itself.

$x - 2 = x - 2$

$x - 2$ occurs $1$ time.

The LCM of $x - 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x - 2$

Multiply each term by $x - 2$ and simplify.

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Multiply each term in $\frac{1}{x - 2} = 0$ by $x - 2$ in order to remove all the denominators from the equation.

$\frac{1}{x - 2} \cdot (x - 2) = 0 \cdot (x - 2)$

Simplify the left side of the equation by cancelling the common factors.

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Cancel the common factor of $x - 2$ .

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Write $x - 2$ as a fraction with denominator $1$ .

$\frac{1}{x - 2} \cdot \frac{x - 2}{1} = 0 \cdot (x - 2)$

Factor out the greatest common factor $x - 2$ .

$\frac{1}{(x - 2) \cdot 1} \cdot \frac{(x - 2) \cdot 1}{1} = 0 \cdot (x - 2)$

Cancel the common factor.

$\frac{1}{(x - 2) \cdot 1} \cdot \frac{(x - 2) \cdot 1}{1} = 0 \cdot (x - 2)$

Rewrite the expression.

$\frac{1}{1} \cdot \frac{1}{1} = 0 \cdot (x - 2)$

Simplify.

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Multiply $\frac{1}{1}$ and $\frac{1}{1}$ .

$\frac{1}{1} = 0 \cdot (x - 2)$

Divide $1$ by $1$ .

$1 = 0 \cdot (x - 2)$

Simplify $0 \cdot (x - 2)$ .

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Multiply $0$ by $x - 2$ .

$1 = 0 (x - 2)$

Multiply $0$ by $x - 2$ .

$1 = 0$

Solve the equation.

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Rewrite the equation as $0 = 1$ .

$0 = 1$

Since $0 \neq 1$ , there are no solutions.

No solution

To find the y-intercept, substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{1}{(0) - 2}$

Solve the equation.

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Remove the extra parentheses.

$y = \frac{1}{0 - 2}$

Simplify $\frac{1}{0 - 2}$ .

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Subtract $2$ from $0$ .

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

Move the negative in front of the fraction.

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

These are the $x$ and $y$ intercepts of the equation $y = \frac{1}{x - 2}$ .

x-intercept: None

y-intercept: $(0, - \frac{1}{2})$

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