Precalculus Examples

$y = \ln (x^{2} - x)$

Step 1

Set the argument of the logarithm equal to zero.

$x^{2} - x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x$ out of $x^{2} - x$ .

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Step 2.1.1

Factor $x$ out of $x^{2}$ .

$x \cdot x - x = 0$

Step 2.1.2

Factor $x$ out of $- x$ .

$x \cdot x + x \cdot - 1 = 0$

Step 2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$x (x - 1) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 1 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.5

The final solution is all the values that make $x (x - 1) = 0$ true.

$x = 0, 1$

Step 3

The vertical asymptote occurs at $x = 0, x = 1$ .

Vertical Asymptote: $x = 0, x = 1$

Step 4