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

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

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

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

Add $3$ to both sides of the equation.

$e^{x - 2} = 3$

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

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

Expand the left side.

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

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

The natural logarithm of $e$ is $1$ .

$(x - 2) \cdot 1 = \ln (3)$

Multiply $x - 2$ by $1$ .

$x - 2 = \ln (3)$

Add $2$ to both sides of the equation.

$x = \ln (3) + 2$

x-intercept(s) in point form.

x-intercept(s): $(\ln (3) + 2, 0)$

Step 2

Find the y-intercepts.

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

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

Simplify each term.

Subtract $2$ from $0$ .

$y = e^{- 2} - 3$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{1}{e^{2}} - 3)$

Step 3

List the intersections.

x-intercept(s): $(\ln (3) + 2, 0)$

y-intercept(s): $(0, \frac{1}{e^{2}} - 3)$

Step 4