Trigonometry Examples

$f (x) = - 10^{x - 1} - 10$

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 $x$ .

$0 = - 10^{x - 1} - 10$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 10^{x - 1} - 10 = 0$ .

$- 10^{x - 1} - 10 = 0$

Step 1.2.2

Add $10$ to both sides of the equation.

$- 10^{x - 1} = 10$

Step 1.2.3

Divide each term in $- 10^{x - 1} = 10$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- 10^{x - 1} = 10$ by $- 1$ .

$\frac{- 10^{x - 1}}{- 1} = \frac{10}{- 1}$

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.

$\frac{10^{x - 1}}{1} = \frac{10}{- 1}$

Step 1.2.3.2.2

Divide $10^{x - 1}$ by $1$ .

$10^{x - 1} = \frac{10}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $10$ by $- 1$ .

$10^{x - 1} = - 10$

Step 1.2.4

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{x - 1}) = \log (- 10)$

Step 1.2.5

The equation cannot be solved because $\log (- 10)$ is undefined.

Undefined

Step 1.2.6

There is no solution for $10^{x - 1} = - 10$

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 $y$ .

$y = - 10^{(0) - 1} - 10$

Step 2.2

Simplify $- 10^{(0) - 1} - 10$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $1$ from $0$ .

$y = - 10^{- 1} - 10$

Step 2.2.1.2

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

$y = - \frac{1}{10} - 10$

Step 2.2.2

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$y = - \frac{1}{10} - 10 \cdot \frac{10}{10}$

Step 2.2.3

Combine $- 10$ and $\frac{10}{10}$ .

$y = - \frac{1}{10} + \frac{- 10 \cdot 10}{10}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{- 1 - 10 \cdot 10}{10}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 10$ by $10$ .

$y = \frac{- 1 - 100}{10}$

Step 2.2.5.2

Subtract $100$ from $- 1$ .

$y = \frac{- 101}{10}$

Step 2.2.6

Move the negative in front of the fraction.

$y = - \frac{101}{10}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{101}{10})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{101}{10})$

Step 4