Calculus Examples

$y = | \ln (x + 1) |$

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 $x$ .

$0 = | \ln (x + 1) |$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $| \ln (x + 1) | = 0$ .

$| \ln (x + 1) | = 0$

Step 1.2.2

Solve for $x$ .

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Step 1.2.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$\ln (x + 1) = \pm 0$

Step 1.2.2.2

Plus or minus $0$ is $0$ .

$\ln (x + 1) = 0$

Step 1.2.2.3

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x + 1)} = e^{0}$

Step 1.2.2.4

Rewrite $\ln (x + 1) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = x + 1$

Step 1.2.2.5

Solve for $x$ .

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Step 1.2.2.5.1

Rewrite the equation as $x + 1 = e^{0}$ .

$x + 1 = e^{0}$

Step 1.2.2.5.2

Anything raised to $0$ is $1$ .

$x + 1 = 1$

Step 1.2.2.5.3

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.2.5.3.1

Subtract $1$ from both sides of the equation.

$x = 1 - 1$

Step 1.2.2.5.3.2

Subtract $1$ from $1$ .

$x = 0$

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 $y$ .

$y = | \ln ((0) + 1) |$

Step 2.2

Simplify $| \ln ((0) + 1) |$ .

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Step 2.2.1

Add $0$ and $1$ .

$y = | \ln (1) |$

Step 2.2.2

The natural logarithm of $1$ is $0$ .

$y = | 0 |$

Step 2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$y = 0$

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