Precalculus Examples

$f (x) = \frac{5 x}{x - 3}$

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 $x$ .

$0 = \frac{5 x}{x - 3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$5 x = 0$

Step 1.2.2

Divide each term in $5 x = 0$ by $5$ and simplify.

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Step 1.2.2.1

Divide each term in $5 x = 0$ by $5$ .

$\frac{5 x}{5} = \frac{0}{5}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{0}{5}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{5}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $5$ .

$x = 0$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0)$

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 $y$ .

$y = \frac{5 (0)}{(0) - 3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{5 (0)}{0 - 3}$

Step 2.2.2

Remove parentheses.

$y = \frac{5 (0)}{(0) - 3}$

Step 2.2.3

Simplify $\frac{5 (0)}{(0) - 3}$ .

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Step 2.2.3.1

Cancel the common factor of $0$ and $(0) - 3$ .

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Step 2.2.3.1.1

Factor $3$ out of $5 (0)$ .

$y = \frac{3 (5 \cdot (0))}{(0) - 3}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $3$ out of $0$ .

$y = \frac{3 (5 \cdot (0))}{3 (0) - 3}$

Step 2.2.3.1.2.2

Factor $3$ out of $- 3$ .

$y = \frac{3 (5 \cdot (0))}{3 \cdot 0 + 3 \cdot - 1}$

Step 2.2.3.1.2.3

Factor $3$ out of $3 \cdot 0 + 3 \cdot - 1$ .

$y = \frac{3 (5 \cdot (0))}{3 \cdot (0 - 1)}$

Step 2.2.3.1.2.4

Cancel the common factor.

$y = \frac{3 (5 \cdot (0))}{3 \cdot (0 - 1)}$

Step 2.2.3.1.2.5

Rewrite the expression.

$y = \frac{5 \cdot (0)}{0 - 1}$

Step 2.2.3.2

Multiply $5$ by $0$ .

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

Step 2.2.3.3

Simplify the expression.

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Step 2.2.3.3.1

Subtract $1$ from $0$ .

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

Step 2.2.3.3.2

Divide $0$ by $- 1$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0)$

y-intercept(s): $(0, 0)$

Step 4