Algebra Examples

$f (x) = - {(\frac{1}{3})}^{x - 1} + 2$

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 $x$ .

$0 = - {(\frac{1}{3})}^{x - 1} + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- {(\frac{1}{3})}^{x - 1} + 2 = 0$ .

$- {(\frac{1}{3})}^{x - 1} + 2 = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Apply the product rule to $\frac{1}{3}$ .

$- \frac{1^{x - 1}}{3^{x - 1}} + 2 = 0$

Step 1.2.2.2

One to any power is one.

$- \frac{1}{3^{x - 1}} + 2 = 0$

Step 1.2.3

Subtract $2$ from both sides of the equation.

$- \frac{1}{3^{x - 1}} = - 2$

Step 1.2.4

Multiply both sides by $3^{x - 1}$ .

$- \frac{1}{3^{x - 1}} \cdot 3^{x - 1} = - 2 \cdot 3^{x - 1}$

Step 1.2.5

Simplify the left side.

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Step 1.2.5.1

Cancel the common factor of $3^{x - 1}$ .

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Step 1.2.5.1.1

Move the leading negative in $- \frac{1}{3^{x - 1}}$ into the numerator.

$\frac{- 1}{3^{x - 1}} \cdot 3^{x - 1} = - 2 \cdot 3^{x - 1}$

Step 1.2.5.1.2

Cancel the common factor.

$\frac{- 1}{3^{x - 1}} \cdot 3^{x - 1} = - 2 \cdot 3^{x - 1}$

Step 1.2.5.1.3

Rewrite the expression.

$- 1 = - 2 \cdot 3^{x - 1}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Rewrite the equation as $- 2 \cdot 3^{x - 1} = - 1$ .

$- 2 \cdot 3^{x - 1} = - 1$

Step 1.2.6.2

Divide each term in $- 2 \cdot 3^{x - 1} = - 1$ by $- 2$ and simplify.

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Step 1.2.6.2.1

Divide each term in $- 2 \cdot 3^{x - 1} = - 1$ by $- 2$ .

$\frac{- 2 \cdot 3^{x - 1}}{- 2} = \frac{- 1}{- 2}$

Step 1.2.6.2.2

Simplify the left side.

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Step 1.2.6.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.6.2.2.1.1

Cancel the common factor.

$\frac{- 2 \cdot 3^{x - 1}}{- 2} = \frac{- 1}{- 2}$

Step 1.2.6.2.2.1.2

Divide $3^{x - 1}$ by $1$ .

$3^{x - 1} = \frac{- 1}{- 2}$

Step 1.2.6.2.3

Simplify the right side.

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Step 1.2.6.2.3.1

Dividing two negative values results in a positive value.

$3^{x - 1} = \frac{1}{2}$

Step 1.2.6.3

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

$\ln (3^{x - 1}) = \ln (\frac{1}{2})$

Step 1.2.6.4

Expand $\ln (3^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x - 1) \ln (3) = \ln (\frac{1}{2})$

Step 1.2.6.5

Simplify the left side.

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Step 1.2.6.5.1

Simplify $(x - 1) \ln (3)$ .

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Step 1.2.6.5.1.1

Apply the distributive property.

$x \ln (3) - 1 \ln (3) = \ln (\frac{1}{2})$

Step 1.2.6.5.1.2

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

$x \ln (3) - \ln (3) = \ln (\frac{1}{2})$

Step 1.2.6.6

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (3) - \ln (3) - \ln (\frac{1}{2}) = 0$

Step 1.2.6.7

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

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Step 1.2.6.7.1

Add $\ln (3)$ to both sides of the equation.

$x \ln (3) - \ln (\frac{1}{2}) = \ln (3)$

Step 1.2.6.7.2

Add $\ln (\frac{1}{2})$ to both sides of the equation.

$x \ln (3) = \ln (3) + \ln (\frac{1}{2})$

Step 1.2.6.8

Divide each term in $x \ln (3) = \ln (3) + \ln (\frac{1}{2})$ by $\ln (3)$ and simplify.

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Step 1.2.6.8.1

Divide each term in $x \ln (3) = \ln (3) + \ln (\frac{1}{2})$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (3)}{\ln (3)} + \frac{\ln (\frac{1}{2})}{\ln (3)}$

Step 1.2.6.8.2

Simplify the left side.

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Step 1.2.6.8.2.1

Cancel the common factor of $\ln (3)$ .

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Step 1.2.6.8.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (3)}{\ln (3)} + \frac{\ln (\frac{1}{2})}{\ln (3)}$

Step 1.2.6.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (\frac{1}{2})}{\ln (3)}$

Step 1.2.6.8.3

Simplify the right side.

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Step 1.2.6.8.3.1

Cancel the common factor of $\ln (3)$ .

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Step 1.2.6.8.3.1.1

Cancel the common factor.

$x = \frac{\ln (3)}{\ln (3)} + \frac{\ln (\frac{1}{2})}{\ln (3)}$

Step 1.2.6.8.3.1.2

Rewrite the expression.

$x = 1 + \frac{\ln (\frac{1}{2})}{\ln (3)}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1 + \frac{\ln (\frac{1}{2})}{\ln (3)}, 0)$

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 $y$ .

$y = - {(\frac{1}{3})}^{(0) - 1} + 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - {(\frac{1}{3})}^{0 - 1} + 2$

Step 2.2.2

Remove parentheses.

$y = - {(\frac{1}{3})}^{(0) - 1} + 2$

Step 2.2.3

Simplify $- {(\frac{1}{3})}^{(0) - 1} + 2$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Subtract $1$ from $0$ .

$y = - {(\frac{1}{3})}^{- 1} + 2$

Step 2.2.3.1.2

Change the sign of the exponent by rewriting the base as its reciprocal.

$y = - 1 \cdot 3 + 2$

Step 2.2.3.1.3

Multiply $- 1$ by $3$ .

$y = - 3 + 2$

Step 2.2.3.2

Add $- 3$ and $2$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(1 + \frac{\ln (\frac{1}{2})}{\ln (3)}, 0)$

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

Step 4