Calculus Examples

$f (x) = x \ln (x)$

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 $x$ .

$0 = x \ln (x)$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$x \ln (x) = 0$

Step 1.2.2

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

$x = 0$

$\ln (x) = 0$

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

Set $\ln (x)$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $\ln (x)$ equal to $0$ .

$\ln (x) = 0$

Step 1.2.4.2

Solve $\ln (x) = 0$ for $x$ .

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Step 1.2.4.2.1

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

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

Step 1.2.4.2.2

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

$e^{0} = x$

Step 1.2.4.2.3

Solve for $x$ .

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Step 1.2.4.2.3.1

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

$x = e^{0}$

Step 1.2.4.2.3.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 1.2.5

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

$x = 0, 1$

Step 1.2.6

Exclude the solutions that do not make $0 = x \ln (x)$ true.

$x = 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0)$

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 $y$ .

$y = (0) \ln (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

The natural logarithm of zero is undefined.

$y = Undefined$

Step 2.2.2

Remove parentheses.

$y = (0) \ln (0)$

Step 2.2.3

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(1, 0)$

y-intercept(s): $None$

Step 4