Finite Math Examples

$\frac{e^{2 x} - 1}{2 x}$

Step 1

Write $\frac{e^{2 x} - 1}{2 x}$ as an equation.

$y = \frac{e^{2 x} - 1}{2 x}$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = \frac{e^{2 x} - 1}{2 x}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

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

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

Add $1$ to both sides of the equation.

$e^{2 x} = 1$

Step 2.2.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 2.2.2.3

Expand the left side.

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Step 2.2.2.3.1

Expand $\ln (e^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (e) = \ln (1)$

Step 2.2.2.3.2

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

$2 x \cdot 1 = \ln (1)$

Step 2.2.2.3.3

Multiply $2$ by $1$ .

$2 x = \ln (1)$

Step 2.2.2.4

Simplify the right side.

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Step 2.2.2.4.1

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

$2 x = 0$

Step 2.2.2.5

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.2.2.5.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2.2.5.2

Simplify the left side.

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Step 2.2.2.5.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.2.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.2.2.5.3

Simplify the right side.

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Step 2.2.2.5.3.1

Divide $0$ by $2$ .

$x = 0$

Step 2.2.3

Exclude the solutions that do not make $0 = \frac{e^{2 x} - 1}{2 x}$ true.

$No solution$

Step 2.3

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = \frac{e^{2 (0)} - 1}{2 (0)}$

Step 3.2

The equation has an undefined fraction.

Undefined

Step 3.3

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

y-intercept(s): $None$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 5