Calculus Examples

$y = \frac{x^{4} + 1}{x^{2} - 16}$

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 $x$ .

$0 = \frac{x^{4} + 1}{x^{2} - 16}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{4} + 1 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $1$ from both sides of the equation.

$x^{4} = - 1$

Step 1.2.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{- 1}$

Step 1.2.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt[4]{- 1}$

Step 1.2.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt[4]{- 1}$

Step 1.2.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt[4]{- 1}, - \sqrt[4]{- 1}$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{x^{4} + 1}{x^{2} - 16}$ 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 $y$ .

$y = \frac{{(0)}^{4} + 1}{{(0)}^{2} - 16}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{4} + 1}{{(0)}^{2} - 16}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{4} + 1}{0^{2} - 16}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{4} + 1}{{(0)}^{2} - 16}$

Step 2.2.4

Simplify $\frac{{(0)}^{4} + 1}{{(0)}^{2} - 16}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{0 + 1}{0^{2} - 16}$

Step 2.2.4.1.2

Add $0$ and $1$ .

$y = \frac{1}{0^{2} - 16}$

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 $0$ .

$y = \frac{1}{0 - 16}$

Step 2.2.4.2.2

Subtract $16$ from $0$ .

$y = \frac{1}{- 16}$

Step 2.2.4.3

Move the negative in front of the fraction.

$y = - \frac{1}{16}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{1}{16})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{1}{16})$

Step 4