Precalculus Examples

$f (x) = - e^{3 x + 1} - 5$

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 $x$ .

$0 = - e^{3 x + 1} - 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- e^{3 x + 1} - 5 = 0$ .

$- e^{3 x + 1} - 5 = 0$

Step 1.2.2

Add $5$ to both sides of the equation.

$- e^{3 x + 1} = 5$

Step 1.2.3

Divide each term in $- e^{3 x + 1} = 5$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- e^{3 x + 1} = 5$ by $- 1$ .

$\frac{- e^{3 x + 1}}{- 1} = \frac{5}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{e^{3 x + 1}}{1} = \frac{5}{- 1}$

Step 1.2.3.2.2

Divide $e^{3 x + 1}$ by $1$ .

$e^{3 x + 1} = \frac{5}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $5$ by $- 1$ .

$e^{3 x + 1} = - 5$

Step 1.2.4

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

$\ln (e^{3 x + 1}) = \ln (- 5)$

Step 1.2.5

The equation cannot be solved because $\ln (- 5)$ is undefined.

Undefined

Step 1.2.6

There is no solution for $e^{3 x + 1} = - 5$

No solution

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y = - e^{3 (0) + 1} - 5$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $3$ by $0$ .

$y = - e^{0 + 1} - 5$

Step 2.2.2

Add $0$ and $1$ .

$y = - e^{1} - 5$

Step 2.2.3

Simplify.

$y = - e - 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - e - 5)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - e - 5)$

Step 4