Algebra Examples

$f (x) = 4 \cdot {(\frac{2}{3})}^{x} + 1$

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 $x$ .

$0 = 4 \cdot {(\frac{2}{3})}^{x} + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$0 = 4 \cdot {(\frac{2}{3})}^{x} + 1$

Step 1.2.2

Simplify $4 \cdot {(\frac{2}{3})}^{x} + 1$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Apply the product rule to $\frac{2}{3}$ .

$0 = 4 \cdot \frac{2^{x}}{3^{x}} + 1$

Step 1.2.2.1.2

Multiply $4 \frac{2^{x}}{3^{x}}$ .

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Step 1.2.2.1.2.1

Combine $4$ and $\frac{2^{x}}{3^{x}}$ .

$0 = \frac{4 \cdot 2^{x}}{3^{x}} + 1$

Step 1.2.2.1.2.2

Rewrite $4$ as $2^{2}$ .

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

Step 1.2.2.1.2.3

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

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

Step 1.2.2.2

Combine into one fraction.

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

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 $y$ .

$y = 4 \cdot {(\frac{2}{3})}^{0} + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot {(\frac{2}{3})}^{0} + 1$

Step 2.2.2

Simplify $4 \cdot {(\frac{2}{3})}^{0} + 1$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Apply the product rule to $\frac{2}{3}$ .

$y = 4 \cdot \frac{2^{0}}{3^{0}} + 1$

Step 2.2.2.1.2

Anything raised to $0$ is $1$ .

$y = 4 \cdot \frac{1}{3^{0}} + 1$

Step 2.2.2.1.3

Anything raised to $0$ is $1$ .

$y = 4 \cdot \frac{1}{1} + 1$

Step 2.2.2.1.4

Cancel the common factor of $1$ .

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Step 2.2.2.1.4.1

Cancel the common factor.

$y = 4 \cdot \frac{1}{1} + 1$

Step 2.2.2.1.4.2

Rewrite the expression.

$y = 4 \cdot 1 + 1$

Step 2.2.2.1.5

Multiply $4$ by $1$ .

$y = 4 + 1$

Step 2.2.2.2

Add $4$ and $1$ .

$y = 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 5)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 5)$

Step 4