Calculus Examples

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

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 $x$ .

$0 = (1 - x) \cdot e^{x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(1 - x) \cdot e^{x} = 0$ .

$(1 - x) \cdot e^{x} = 0$

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 $0$ .

$1 - x = 0$

$e^{x} = 0$

Step 1.2.3

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 1.2.3.2

Solve $1 - x = 0$ for $x$ .

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Step 1.2.3.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

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

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

Step 1.2.3.2.2.2

Simplify the left side.

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Step 1.2.3.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2.2.2

Divide $x$ by $1$ .

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

Step 1.2.3.2.2.3

Simplify the right side.

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Step 1.2.3.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 1.2.4

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

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

$e^{x} = 0$

Step 1.2.4.2

Solve $e^{x} = 0$ for $x$ .

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Step 1.2.4.2.1

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

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

Step 1.2.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 1.2.4.2.3

There is no solution for $e^{x} = 0$

No solution

Step 1.2.5

The final solution is all the values that make $(1 - x) \cdot e^{x} = 0$ 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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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Step 2.2.2.1

Subtract $0$ from $1$ .

$y = 1 \cdot e^{0}$

Step 2.2.2.2

Multiply $e^{0}$ by $1$ .

$y = e^{0}$

Step 2.2.2.3

Anything raised to $0$ is $1$ .

$y = 1$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4