Precalculus Examples

$f (x) = 1 - e^{- x}$

Step 1

Set $1 - e^{- x}$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Subtract $1$ from both sides of the equation.

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

Step 2.2

Divide each term in $- e^{- x} = - 1$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- e^{- x} = - 1$ by $- 1$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $e^{- x}$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$e^{- x} = 1$

Step 2.3

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

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

Step 2.4

Expand the left side.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Multiply $- 1$ by $1$ .

$- x = \ln (1)$

Step 2.5

Simplify the right side.

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Step 2.5.1

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

$- x = 0$

Step 2.6

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 2.6.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 2.6.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 3