Find the x and y Intercepts f(x)=(1/2)^x-3

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

Simplify each term.

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

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

One to any power is one.

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

Add $3$ to both sides of the equation.

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

Multiply both sides by $2^{x}$ .

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

Simplify the left side.

Cancel the common factor of $2^{x}$ .

Cancel the common factor.

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

Rewrite the expression.

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

Solve for $x$ .

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $2^{x}$ by $1$ .

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

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

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

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

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

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

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

Simplify each term.

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

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

Anything raised to $0$ is $1$ .

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

Anything raised to $0$ is $1$ .

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

Divide $1$ by $1$ .

$y = 1 - 3$

Subtract $3$ from $1$ .

$y = - 2$

y-intercept(s) in point form.

y-intercept(s): $(0, - 2)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 2)$

Step 4

Find the x and y Intercepts f(x)=(1/2)^x-3

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