Calculus Examples

$G (x) = 3 (x + 1) e^{- 2 x}$

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 $x$ .

$0 = 3 (x + 1) \cdot e^{- 2 x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 (x + 1) \cdot e^{- 2 x} = 0$ .

$3 (x + 1) \cdot e^{- 2 x} = 0$

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 $0$ .

$x + 1 = 0$

$e^{- 2 x} = 0$

Step 1.2.3

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

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Step 1.2.3.1

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

$x + 1 = 0$

Step 1.2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.4

Set $e^{- 2 x}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $e^{- 2 x}$ equal to $0$ .

$e^{- 2 x} = 0$

Step 1.2.4.2

Solve $e^{- 2 x} = 0$ 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.

$\ln (e^{- 2 x}) = \ln (0)$

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 $e^{- 2 x} = 0$

No solution

Step 1.2.5

The final solution is all the values that make $3 (x + 1) \cdot e^{- 2 x} = 0$ 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 $y$ .

$y = 3 ((0) + 1) \cdot e^{- 2 \cdot 0}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 (0 + 1) \cdot e^{- 2 \cdot 0}$

Step 2.2.2

Remove parentheses.

$y = 3 ((0) + 1) \cdot e^{- 2 \cdot 0}$

Step 2.2.3

Simplify $3 ((0) + 1) \cdot e^{- 2 \cdot 0}$ .

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Step 2.2.3.1

Add $0$ and $1$ .

$y = 3 \cdot 1 \cdot e^{- 2 \cdot 0}$

Step 2.2.3.2

Multiply $3$ by $1$ .

$y = 3 \cdot e^{- 2 \cdot 0}$

Step 2.2.3.3

Multiply $- 2$ by $0$ .

$y = 3 \cdot e^{0}$

Step 2.2.3.4

Anything raised to $0$ is $1$ .

$y = 3 \cdot 1$

Step 2.2.3.5

Multiply $3$ by $1$ .

$y = 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3)$

Step 3

List the intersections.

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

y-intercept(s): $(0, 3)$

Step 4