Algebra Examples

$F (x) = 3^{x - 2} + 1$

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 $x$ .

$0 = 3^{x - 2} + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3^{x - 2} + 1 = 0$ .

$3^{x - 2} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$3^{x - 2} = - 1$

Step 1.2.3

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

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

Step 1.2.4

The equation cannot be solved because $\ln (- 1)$ is undefined.

Undefined

Step 1.2.5

There is no solution for $3^{x - 2} = - 1$

No solution

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

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

Step 2.2

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

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Subtract $2$ from $0$ .

$y = 3^{- 2} + 1$

Step 2.2.1.2

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

$y = \frac{1}{3^{2}} + 1$

Step 2.2.1.3

Raise $3$ to the power of $2$ .

$y = \frac{1}{9} + 1$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Write $1$ as a fraction with a common denominator.

$y = \frac{1}{9} + \frac{9}{9}$

Step 2.2.2.2

Combine the numerators over the common denominator.

$y = \frac{1 + 9}{9}$

Step 2.2.2.3

Add $1$ and $9$ .

$y = \frac{10}{9}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{10}{9})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \frac{10}{9})$

Step 4