Precalculus Examples

$f (x) = \frac{x - 2}{x^{2} + x - 6}$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = \frac{x - 2}{x^{2} + x - 6}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x - 2 = 0$

Step 1.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x - 2}{x^{2} + x - 6}$ true.

$No solution$

Step 1.3

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

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = \frac{(0) - 2}{{(0)}^{2} + 0 - 6}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{(0) - 2}{{(0)}^{2} + 0 - 6}$

Step 2.2.2

Remove parentheses.

$y = \frac{0 - 2}{{(0)}^{2} + 0 - 6}$

Step 2.2.3

Remove parentheses.

$y = \frac{0 - 2}{0^{2} + 0 - 6}$

Step 2.2.4

Remove parentheses.

$y = \frac{(0) - 2}{{(0)}^{2} + 0 - 6}$

Step 2.2.5

Simplify $\frac{(0) - 2}{{(0)}^{2} + 0 - 6}$ .

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Step 2.2.5.1

Subtract $2$ from $0$ .

$y = \frac{- 2}{0^{2} + 0 - 6}$

Step 2.2.5.2

Simplify the denominator.

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Step 2.2.5.2.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{- 2}{0 + 0 - 6}$

Step 2.2.5.2.2

Add $0$ and $0$ .

$y = \frac{- 2}{0 - 6}$

Step 2.2.5.2.3

Subtract $6$ from $0$ .

$y = \frac{- 2}{- 6}$

Step 2.2.5.3

Cancel the common factor of $- 2$ and $- 6$ .

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Step 2.2.5.3.1

Factor $- 2$ out of $- 2$ .

$y = \frac{- 2 (1)}{- 6}$

Step 2.2.5.3.2

Cancel the common factors.

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Step 2.2.5.3.2.1

Factor $- 2$ out of $- 6$ .

$y = \frac{- 2 \cdot 1}{- 2 \cdot 3}$

Step 2.2.5.3.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 1}{- 2 \cdot 3}$

Step 2.2.5.3.2.3

Rewrite the expression.

$y = \frac{1}{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{3})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \frac{1}{3})$

Step 4