Algebra Examples

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

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 $x$ .

$0 = - 5^{x - 5} - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$- 5^{x - 5} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$- 5^{x - 5} = 1$

Step 1.2.3

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

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Step 1.2.3.1

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

$\frac{- 5^{x - 5}}{- 1} = \frac{1}{- 1}$

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.

$\frac{5^{x - 5}}{1} = \frac{1}{- 1}$

Step 1.2.3.2.2

Divide $5^{x - 5}$ by $1$ .

$5^{x - 5} = \frac{1}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $1$ by $- 1$ .

$5^{x - 5} = - 1$

Step 1.2.4

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

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

Step 1.2.5

The equation cannot be solved because $\ln (- 1)$ is undefined.

Undefined

Step 1.2.6

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

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 $y$ .

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

Step 2.2

Simplify $- 5^{(0) - 5} - 1$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $5$ from $0$ .

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

Step 2.2.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = - \frac{1}{5^{5}} - 1$

Step 2.2.1.3

Raise $5$ to the power of $5$ .

$y = - \frac{1}{3125} - 1$

Step 2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3125}{3125}$ .

$y = - \frac{1}{3125} - 1 \cdot \frac{3125}{3125}$

Step 2.2.3

Combine $- 1$ and $\frac{3125}{3125}$ .

$y = - \frac{1}{3125} + \frac{- 1 \cdot 3125}{3125}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{- 1 - 1 \cdot 3125}{3125}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 1$ by $3125$ .

$y = \frac{- 1 - 3125}{3125}$

Step 2.2.5.2

Subtract $3125$ from $- 1$ .

$y = \frac{- 3126}{3125}$

Step 2.2.6

Move the negative in front of the fraction.

$y = - \frac{3126}{3125}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{3126}{3125})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{3126}{3125})$

Step 4