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

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

Step 1

Find the x-intercepts.

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

$0 = 2^{- x} - 3$

Solve the equation.

Rewrite the equation as $2^{- x} - 3 = 0$ .

$2^{- x} - 3 = 0$

Add $3$ to both sides of the equation.

$2^{- x} = 3$

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

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

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

$- x \ln (2) = \ln (3)$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

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

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

Simplify each term.

Multiply $- 1$ by $0$ .

$y = 2^{0} - 3$

Anything raised to $0$ is $1$ .

$y = 1 - 3$

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): $(- \frac{\ln (3)}{\ln (2)}, 0)$

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

Step 4

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

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