Precalculus Examples

${(\frac{1}{3})}^{x}$

Step 1

Write ${(\frac{1}{3})}^{x}$ as an equation.

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

Step 2

Find the x-intercepts.

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Step 2.1

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

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

Step 2.2

Solve the equation.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.2.4

There is no solution for ${(\frac{1}{3})}^{x} = 0$

No solution

Step 2.3

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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Step 3.1

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

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

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Anything raised to $0$ is $1$ .

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

Step 3.2.2.3

Anything raised to $0$ is $1$ .

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

Step 3.2.2.4

Divide $1$ by $1$ .

$y = 1$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $None$

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

Step 5