Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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

Step 1.2.2.2

One to any power is one.

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

Step 1.2.3

Add $1$ to both sides of the equation.

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

Step 1.2.4

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

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

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

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

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Step 1.2.5.1.1.1

Cancel the common factor.

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

Step 1.2.5.1.1.2

Rewrite the expression.

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

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Multiply $3^{x}$ by $1$ .

$1 = 3^{x}$

Step 1.2.6

Solve for $x$ .

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Step 1.2.6.1

Rewrite the equation as $3^{x} = 1$ .

$3^{x} = 1$

Step 1.2.6.2

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

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

Step 1.2.6.3

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

$x \ln (3) = \ln (1)$

Step 1.2.6.4

Simplify the right side.

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Step 1.2.6.4.1

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

$x \ln (3) = 0$

Step 1.2.6.5

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ and simplify.

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Step 1.2.6.5.1

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ .

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

Step 1.2.6.5.2

Simplify the left side.

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Step 1.2.6.5.2.1

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

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Step 1.2.6.5.2.1.1

Cancel the common factor.

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

Step 1.2.6.5.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.6.5.3

Simplify the right side.

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Step 1.2.6.5.3.1

Divide $0$ by $\ln (3)$ .

$x = 0$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$y = \frac{1^{0}}{3^{0}} - 1$

Step 2.2.2.1.2

Anything raised to $0$ is $1$ .

$y = \frac{1}{3^{0}} - 1$

Step 2.2.2.1.3

Anything raised to $0$ is $1$ .

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

Step 2.2.2.1.4

Divide $1$ by $1$ .

$y = 1 - 1$

Step 2.2.2.2

Subtract $1$ from $1$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4