Find the x and y Intercepts f(x)=3(1/2)^(x-1)+2

$f (x) = 3 {(\frac{1}{2})}^{x - 1} + 2$

Step 1

Find the x-intercepts.

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

$0 = 3 {(\frac{1}{2})}^{x - 1} + 2$

Solve the equation.

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

$3 {(\frac{1}{2})}^{x - 1} + 2 = 0$

Simplify each term.

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

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

One to any power is one.

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

Combine $3$ and $\frac{1}{2^{x - 1}}$ .

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

Subtract $2$ from both sides of the equation.

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

Multiply both sides by $2^{x - 1}$ .

$\frac{3}{2^{x - 1}} \cdot 2^{x - 1} = - 2 \cdot 2^{x - 1}$

Simplify.

Simplify the left side.

Cancel the common factor of $2^{x - 1}$ .

Cancel the common factor.

$\frac{3}{2^{x - 1}} \cdot 2^{x - 1} = - 2 \cdot 2^{x - 1}$

Rewrite the expression.

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

Simplify the right side.

Multiply $- 2 \cdot 2^{x - 1}$ .

Factor out negative.

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

Raise $2$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Subtract $1$ from $1$ .

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

Add $x$ and $0$ .

$3 = - 2^{x}$

Solve for $x$ .

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

$- 2^{x} = 3$

Divide each term in $- 2^{x} = 3$ by $- 1$ and simplify.

Divide each term in $- 2^{x} = 3$ by $- 1$ .

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Divide $3$ by $- 1$ .

$2^{x} = - 3$

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

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

The equation cannot be solved because $\ln (- 3)$ is undefined.

Undefined

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

No solution

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.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

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

Simplify each term.

Subtract $1$ from $0$ .

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

Change the sign of the exponent by rewriting the base as its reciprocal.

$y = 3 \cdot 2 + 2$

Multiply $3$ by $2$ .

$y = 6 + 2$

Add $6$ and $2$ .

$y = 8$

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 8)$

Step 4

Find the x and y Intercepts f(x)=3(1/2)^(x-1)+2

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