Precalculus Examples

$f (x) = 100 - 10^{0.1 x + 1}$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$0 = 100 - 10^{0.1 x + 1}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $100 - 10^{0.1 x + 1} = 0$ .

$100 - 10^{0.1 x + 1} = 0$

Step 1.2.2

Subtract $100$ from both sides of the equation.

$- 10^{0.1 x + 1} = - 100$

Step 1.2.3

Divide each term in $- 10^{0.1 x + 1} = - 100$ by $- 1$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $- 10^{0.1 x + 1} = - 100$ by $- 1$ .

$\frac{- 10^{0.1 x + 1}}{- 1} = \frac{- 100}{- 1}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{10^{0.1 x + 1}}{1} = \frac{- 100}{- 1}$

Step 1.2.3.2.2

Divide $10^{0.1 x + 1}$ by $1$ .

$10^{0.1 x + 1} = \frac{- 100}{- 1}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Divide $- 100$ by $- 1$ .

$10^{0.1 x + 1} = 100$

Step 1.2.4

Create equivalent expressions in the equation that all have equal bases.

$10^{0.1 x + 1} = 10^{2}$

Step 1.2.5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$0.1 x + 1 = 2$

Step 1.2.6

Solve for $x$ .

Tap for more steps...

Step 1.2.6.1

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 1.2.6.1.1

Subtract $1$ from both sides of the equation.

$0.1 x = 2 - 1$

Step 1.2.6.1.2

Subtract $1$ from $2$ .

$0.1 x = 1$

Step 1.2.6.2

Divide each term in $0.1 x = 1$ by $0.1$ and simplify.

Tap for more steps...

Step 1.2.6.2.1

Divide each term in $0.1 x = 1$ by $0.1$ .

$\frac{0.1 x}{0.1} = \frac{1}{0.1}$

Step 1.2.6.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.6.2.2.1

Cancel the common factor of $0.1$ .

Tap for more steps...

Step 1.2.6.2.2.1.1

Cancel the common factor.

$\frac{0.1 x}{0.1} = \frac{1}{0.1}$

Step 1.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{0.1}$

Step 1.2.6.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.6.2.3.1

Divide $1$ by $0.1$ .

$x = 10$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(10, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

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

$y = 100 - 10^{0.1 (0) + 1}$

Step 2.2

Simplify $100 - 10^{0.1 (0) + 1}$ .

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply $0.1$ by $0$ .

$y = 100 - 10^{0 + 1}$

Step 2.2.1.2

Add $0$ and $1$ .

$y = 100 - 10^{1}$

Step 2.2.1.3

Evaluate the exponent.

$y = 100 - 1 \cdot 10$

Step 2.2.1.4

Multiply $- 1$ by $10$ .

$y = 100 - 10$

Step 2.2.2

Subtract $10$ from $100$ .

$y = 90$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 90)$

Step 3

List the intersections.

x-intercept(s): $(10, 0)$

y-intercept(s): $(0, 90)$

Step 4