Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

Add $3$ to both sides of the equation.

$e^{x - 2} = 3$

Step 1.2.3

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

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

Step 1.2.4

Expand the left side.

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

Multiply $x - 2$ by $1$ .

$x - 2 = \ln (3)$

Step 1.2.5

Add $2$ to both sides of the equation.

$x = \ln (3) + 2$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Subtract $2$ from $0$ .

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

Step 2.2.2

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

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

Step 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