Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Combine $\frac{1}{3}$ and $x$ .

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

Step 1.2.2.2

Multiply $- 2 \cdot 2^{\frac{x}{3}}$ .

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Step 1.2.2.2.1

Factor out negative.

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

Step 1.2.2.2.2

Raise $2$ to the power of $1$ .

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

Step 1.2.2.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.3

Subtract $3$ from both sides of the equation.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.4.2.2

Divide $2^{1 + \frac{x}{3}}$ by $1$ .

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $- 3$ by $- 1$ .

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

Step 1.2.5

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

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

Step 1.2.6

Expand $\ln (2^{1 + \frac{x}{3}})$ by moving $1 + \frac{x}{3}$ outside the logarithm.

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

Step 1.2.7

Simplify the left side.

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Step 1.2.7.1

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

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Step 1.2.7.1.1

Apply the distributive property.

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

Step 1.2.7.1.2

Multiply $\ln (2)$ by $1$ .

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

Step 1.2.7.1.3

Combine $\frac{x}{3}$ and $\ln (2)$ .

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

Step 1.2.8

Reorder $\ln (2)$ and $\frac{x \ln (2)}{3}$ .

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

Step 1.2.9

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

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

Step 1.2.10

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 1.2.11

Subtract $\ln (\frac{2}{3})$ from both sides of the equation.

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

Step 1.2.12

Multiply both sides of the equation by $3$ .

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

Step 1.2.13

Simplify both sides of the equation.

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Step 1.2.13.1

Simplify the left side.

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Step 1.2.13.1.1

Cancel the common factor of $3$ .

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Step 1.2.13.1.1.1

Cancel the common factor.

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

Step 1.2.13.1.1.2

Rewrite the expression.

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

Step 1.2.13.2

Simplify the right side.

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Step 1.2.13.2.1

Multiply $- 1$ by $3$ .

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

Step 1.2.14

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

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Step 1.2.14.1

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

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

Step 1.2.14.2

Simplify the left side.

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Step 1.2.14.2.1

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

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Step 1.2.14.2.1.1

Cancel the common factor.

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

Step 1.2.14.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.14.3

Simplify the right side.

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Step 1.2.14.3.1

Move the negative in front of the fraction.

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

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

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

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $\frac{1}{3}$ by $0$ .

$y = - 2 \cdot 2^{0} + 3$

Step 2.2.1.2

Anything raised to $0$ is $1$ .

$y = - 2 \cdot 1 + 3$

Step 2.2.1.3

Multiply $- 2$ by $1$ .

$y = - 2 + 3$

Step 2.2.2

Add $- 2$ and $3$ .

$y = 1$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4