Find the x and y Intercepts f(x)=(1/2)^(-x)-5

$f (x) = {(\frac{1}{2})}^{- x} - 5$

Step 1

Find the x-intercepts.

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

$0 = {(\frac{1}{2})}^{- x} - 5$

Solve the equation.

Rewrite the equation as ${(\frac{1}{2})}^{- x} - 5 = 0$ .

${(\frac{1}{2})}^{- x} - 5 = 0$

Simplify each term.

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

$\frac{1^{- x}}{2^{- x}} - 5 = 0$

One to any power is one.

$\frac{1}{2^{- x}} - 5 = 0$

Add $5$ to both sides of the equation.

$\frac{1}{2^{- x}} = 5$

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

$\frac{1}{2^{- x}} \cdot 2^{- x} = 5 \cdot 2^{- x}$

Simplify the left side.

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

Cancel the common factor.

$\frac{1}{2^{- x}} \cdot 2^{- x} = 5 \cdot 2^{- x}$

Rewrite the expression.

$1 = 5 \cdot 2^{- x}$

Solve for $x$ .

Rewrite the equation as $5 \cdot 2^{- x} = 1$ .

$5 \cdot 2^{- x} = 1$

Divide each term in $(5 \cdot 2^{- x}) = 1$ by $5$ and simplify.

Divide each term in $(5 \cdot 2^{- x}) = 1$ by $5$ .

$\frac{(5 \cdot 2^{- x})}{5} = \frac{1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 \cdot 2^{- x}}{5} = \frac{1}{5}$

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

$2^{- x} = \frac{1}{5}$

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

$\ln (2^{- x}) = \ln (\frac{1}{5})$

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

$- x \ln (2) = \ln (\frac{1}{5})$

Divide each term in $- x \ln (2) = \ln (\frac{1}{5})$ by $- 1 \ln (2)$ and simplify.

Divide each term in $- x \ln (2) = \ln (\frac{1}{5})$ by $- 1 \ln (2)$ .

$\frac{- x \ln (2)}{- 1 \ln (2)} = \frac{\ln (\frac{1}{5})}{- 1 \ln (2)}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (\frac{1}{5})}{- 1 \ln (2)}$

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

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (\frac{1}{5})}{- 1 \ln (2)}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{5})}{- 1 \ln (2)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{5})}{\ln (2)}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{\ln (\frac{1}{5})}{\ln (2)}, 0)$

Step 2

Find the y-intercepts.

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

$y = {(\frac{1}{2})}^{- (0)} - 5$

Solve the equation.

Remove parentheses.

$y = {(\frac{1}{2})}^{- (0)} - 5$

Simplify ${(\frac{1}{2})}^{- (0)} - 5$ .

Simplify each term.

Multiply $- 1$ by $0$ .

$y = {(\frac{1}{2})}^{0} - 5$

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

$y = \frac{1^{0}}{2^{0}} - 5$

Anything raised to $0$ is $1$ .

$y = \frac{1}{2^{0}} - 5$

Anything raised to $0$ is $1$ .

$y = \frac{1}{1} - 5$

Divide $1$ by $1$ .

$y = 1 - 5$

Subtract $5$ from $1$ .

$y = - 4$

y-intercept(s) in point form.

y-intercept(s): $(0, - 4)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{\ln (\frac{1}{5})}{\ln (2)}, 0)$

y-intercept(s): $(0, - 4)$

Step 4

Find the x and y Intercepts f(x)=(1/2)^(-x)-5

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