Calculus Examples

$f (x) = \frac{x^{2} - 7}{x - 4}$

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 $x$ .

$0 = \frac{x^{2} - 7}{x - 4}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 7 = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Add $7$ to both sides of the equation.

$x^{2} = 7$

Step 1.2.2.2

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

$x = \pm \sqrt{7}$

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.

$x = \sqrt{7}$

Step 1.2.2.3.2

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

$x = - \sqrt{7}$

Step 1.2.2.3.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{7}, 0), (- \sqrt{7}, 0)$

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 $y$ .

$y = \frac{{(0)}^{2} - 7}{(0) - 4}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} - 7}{(0) - 4}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} - 7}{0 - 4}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 7}{(0) - 4}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} - 7}{(0) - 4}$ .

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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 $0$ .

$y = \frac{0 - 7}{0 - 4}$

Step 2.2.4.1.2

Subtract $7$ from $0$ .

$y = \frac{- 7}{0 - 4}$

Step 2.2.4.2

Reduce the expression by cancelling the common factors.

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Step 2.2.4.2.1

Subtract $4$ from $0$ .

$y = \frac{- 7}{- 4}$

Step 2.2.4.2.2

Dividing two negative values results in a positive value.

$y = \frac{7}{4}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{7}{4})$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{7}, 0), (- \sqrt{7}, 0)$

y-intercept(s): $(0, \frac{7}{4})$

Step 4