Precalculus Examples

$y = 2^{x - 1} - 1$

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 $x$ .

$0 = 2^{x - 1} - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2^{x - 1} - 1 = 0$ .

$2^{x - 1} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$2^{x - 1} = 1$

Step 1.2.3

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

$\ln (2^{x - 1}) = \ln (1)$

Step 1.2.4

Expand $\ln (2^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x - 1) \ln (2) = \ln (1)$

Step 1.2.5

Simplify the left side.

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Step 1.2.5.1

Simplify $(x - 1) \ln (2)$ .

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Step 1.2.5.1.1

Apply the distributive property.

$x \ln (2) - 1 \ln (2) = \ln (1)$

Step 1.2.5.1.2

Rewrite $- 1 \ln (2)$ as $- \ln (2)$ .

$x \ln (2) - \ln (2) = \ln (1)$

Step 1.2.6

Simplify the right side.

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Step 1.2.6.1

The natural logarithm of $1$ is $0$ .

$x \ln (2) - \ln (2) = 0$

Step 1.2.7

Add $\ln (2)$ to both sides of the equation.

$x \ln (2) = \ln (2)$

Step 1.2.8

Divide each term in $x \ln (2) = \ln (2)$ by $\ln (2)$ and simplify.

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Step 1.2.8.1

Divide each term in $x \ln (2) = \ln (2)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (2)}{\ln (2)}$

Step 1.2.8.2

Simplify the left side.

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Step 1.2.8.2.1

Cancel the common factor of $\ln (2)$ .

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Step 1.2.8.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (2)}{\ln (2)}$

Step 1.2.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (2)}{\ln (2)}$

Step 1.2.8.3

Simplify the right side.

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Step 1.2.8.3.1

Cancel the common factor of $\ln (2)$ .

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Step 1.2.8.3.1.1

Cancel the common factor.

$x = \frac{\ln (2)}{\ln (2)}$

Step 1.2.8.3.1.2

Rewrite the expression.

$x = 1$

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 $y$ .

$y = 2^{(0) - 1} - 1$

Step 2.2

Simplify $2^{(0) - 1} - 1$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $1$ from $0$ .

$y = 2^{- 1} - 1$

Step 2.2.1.2

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

$y = \frac{1}{2} - 1$

Step 2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{1}{2} - 1 \cdot \frac{2}{2}$

Step 2.2.3

Combine $- 1$ and $\frac{2}{2}$ .

$y = \frac{1}{2} + \frac{- 1 \cdot 2}{2}$

Step 2.2.4

Combine the numerators over the common denominator.

$y = \frac{1 - 1 \cdot 2}{2}$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 1$ by $2$ .

$y = \frac{1 - 2}{2}$

Step 2.2.5.2

Subtract $2$ from $1$ .

$y = \frac{- 1}{2}$

Step 2.2.6

Move the negative in front of the fraction.

$y = - \frac{1}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{1}{2})$

Step 3

List the intersections.

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

y-intercept(s): $(0, - \frac{1}{2})$

Step 4