Calculus Examples

$f (x) = \ln (x^{2} + 1)$

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 $x$ .

$0 = \ln (x^{2} + 1)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\ln (x^{2} + 1) = 0$ .

$\ln (x^{2} + 1) = 0$

Step 1.2.2

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

$e^{\ln (x^{2} + 1)} = e^{0}$

Step 1.2.3

Rewrite $\ln (x^{2} + 1) = 0$ 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 $b^{y} = x$ .

$e^{0} = x^{2} + 1$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $x^{2} + 1 = e^{0}$ .

$x^{2} + 1 = e^{0}$

Step 1.2.4.2

Anything raised to $0$ is $1$ .

$x^{2} + 1 = 1$

Step 1.2.4.3

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

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Step 1.2.4.3.1

Subtract $1$ from both sides of the equation.

$x^{2} = 1 - 1$

Step 1.2.4.3.2

Subtract $1$ from $1$ .

$x^{2} = 0$

Step 1.2.4.4

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

$x = \pm \sqrt{0}$

Step 1.2.4.5

Simplify $\pm \sqrt{0}$ .

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Step 1.2.4.5.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.4.5.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.4.5.3

Plus or minus $0$ is $0$ .

$x = 0$

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 $y$ .

$y = \ln ({(0)}^{2} + 1)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \ln (0^{2} + 1)$

Step 2.2.2

Remove parentheses.

$y = \ln ({(0)}^{2} + 1)$

Step 2.2.3

Simplify $\ln ({(0)}^{2} + 1)$ .

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Step 2.2.3.1

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

$y = \ln (0 + 1)$

Step 2.2.3.2

Add $0$ and $1$ .

$y = \ln (1)$

Step 2.2.3.3

The natural logarithm of $1$ 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