Precalculus Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

Undefined

Step 1.2.4

There is no solution for ${(\frac{1}{2})}^{x + 3} = 0$

No solution

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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Step 2.2.3.1

Add $0$ and $3$ .

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

Step 2.2.3.2

Apply the product rule to $\frac{1}{2}$ .

$y = \frac{1^{3}}{2^{3}}$

Step 2.2.3.3

One to any power is one.

$y = \frac{1}{2^{3}}$

Step 2.2.3.4

Raise $2$ to the power of $3$ .

$y = \frac{1}{8}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{8})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \frac{1}{8})$

Step 4