Precalculus Examples

$f (x) = \frac{x + 5}{x^{2} - 25}$

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 $x$ .

$0 = \frac{x + 5}{x^{2} - 25}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x + 5 = 0$

Step 1.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x + 5}{x^{2} - 25}$ 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 $y$ .

$y = \frac{(0) + 5}{{(0)}^{2} - 25}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0 + 5}{{(0)}^{2} - 25}$

Step 2.2.2

Remove parentheses.

$y = \frac{0 + 5}{0^{2} - 25}$

Step 2.2.3

Remove parentheses.

$y = \frac{(0) + 5}{{(0)}^{2} - 25}$

Step 2.2.4

Simplify $\frac{(0) + 5}{{(0)}^{2} - 25}$ .

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Step 2.2.4.1

Add $0$ and $5$ .

$y = \frac{5}{0^{2} - 25}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{5}{0 - 25}$

Step 2.2.4.2.2

Subtract $25$ from $0$ .

$y = \frac{5}{- 25}$

Step 2.2.4.3

Reduce the expression by cancelling the common factors.

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Step 2.2.4.3.1

Cancel the common factor of $5$ and $- 25$ .

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Step 2.2.4.3.1.1

Factor $5$ out of $5$ .

$y = \frac{5 (1)}{- 25}$

Step 2.2.4.3.1.2

Cancel the common factors.

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Step 2.2.4.3.1.2.1

Factor $5$ out of $- 25$ .

$y = \frac{5 \cdot 1}{5 \cdot - 5}$

Step 2.2.4.3.1.2.2

Cancel the common factor.

$y = \frac{5 \cdot 1}{5 \cdot - 5}$

Step 2.2.4.3.1.2.3

Rewrite the expression.

$y = \frac{1}{- 5}$

Step 2.2.4.3.2

Move the negative in front of the fraction.

$y = - \frac{1}{5}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{1}{5})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{1}{5})$

Step 4