Algebra Examples

$f (x) = \frac{1}{2} \cdot e^{- x} - 1$

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

$x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as

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

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

Step 1.2.2

Combine

$\frac{1}{2}$ and

$e^{- x}$ .

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

Step 1.2.3

Add

$1$ to both sides of the equation.

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

Step 1.2.4

Multiply both sides by

$2$ .

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

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Cancel the common factor of

$2$ .

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Step 1.2.5.1.1.1

Cancel the common factor.

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

Step 1.2.5.1.1.2

Rewrite the expression.

$e^{- x} = 1 \cdot 2$

$e^{- x} = 1 \cdot 2$

$e^{- x} = 1 \cdot 2$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply

$2$ by

$1$ .

$e^{- x} = 2$

$e^{- x} = 2$

$e^{- x} = 2$

Step 1.2.6

Solve for

$x$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

Expand the left side.

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Step 1.2.6.2.1

Expand

$\ln (e^{- x})$ by moving

$- x$ outside the logarithm.

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

Step 1.2.6.2.2

The natural logarithm of

$e$ is

$1$ .

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

Step 1.2.6.2.3

Multiply

$- 1$ by

$1$ .

$- x = \ln (2)$

$- x = \ln (2)$

Step 1.2.6.3

Divide each term in

$- x = \ln (2)$ by

$- 1$ and simplify.

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Step 1.2.6.3.1

Divide each term in

$- x = \ln (2)$ by

$- 1$ .

$\frac{- x}{- 1} = \frac{\ln (2)}{- 1}$

Step 1.2.6.3.2

Simplify the left side.

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Step 1.2.6.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\ln (2)}{- 1}$

Step 1.2.6.3.2.2

Divide

$x$ by

$1$ .

$x = \frac{\ln (2)}{- 1}$

$x = \frac{\ln (2)}{- 1}$

Step 1.2.6.3.3

Simplify the right side.

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Step 1.2.6.3.3.1

Move the negative one from the denominator of

$\frac{\ln (2)}{- 1}$ .

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

Step 1.2.6.3.3.2

Rewrite

$- 1 \cdot \ln (2)$ as

$- \ln (2)$ .

$x = - \ln (2)$

$x = - \ln (2)$

$x = - \ln (2)$

$x = - \ln (2)$

$x = - \ln (2)$

Step 1.3

x-intercept(s) in point form.

x-intercept(s):

$(- \ln (2), 0)$

x-intercept(s):

$(- \ln (2), 0)$

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

$y$ .

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

Step 2.2

Simplify

$\frac{1}{2} \cdot e^{- (0)} - 1$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply

$- 1$ by

$0$ .

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

Step 2.2.1.2

Anything raised to

$0$ is

$1$ .

$y = \frac{1}{2} \cdot 1 - 1$

Step 2.2.1.3

Multiply

$\frac{1}{2}$ by

$1$ .

$y = \frac{1}{2} - 1$

$y = \frac{1}{2} - 1$

Step 2.2.2

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$y = \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 2.2.3

Combine

$- 1$ and

$\frac{2}{2}$ .

$y = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{1 - 1 \cdot 2}{2}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply

$- 1$ by

$2$ .

$y = \frac{1 - 2}{2}$

Step 2.2.5.2

Subtract

$2$ from

$1$ .

$y = \frac{- 1}{2}$

$y = \frac{- 1}{2}$

Step 2.2.6

Move the negative in front of the fraction.

$y = - \frac{1}{2}$

$y = - \frac{1}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s):

$(0, - \frac{1}{2})$

y-intercept(s):

$(0, - \frac{1}{2})$

Step 3

List the intersections.

x-intercept(s):

$(- \ln (2), 0)$

y-intercept(s):

$(0, - \frac{1}{2})$

Step 4

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$