Algebra Examples

$f (x) = \frac{4}{3} x - 2$

Step 1

Write $f (x) = \frac{4}{3} x - 2$ as an equation.

$y = \frac{4}{3} x - 2$

Step 2

Interchange the variables.

$x = \frac{4}{3} y - 2$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{4}{3} y - 2 = x$ .

$\frac{4}{3} y - 2 = x$

Step 3.2

Combine $\frac{4}{3}$ and $y$ .

$\frac{4 y}{3} - 2 = x$

Step 3.3

Add $2$ to both sides of the equation.

$\frac{4 y}{3} = x + 2$

Step 3.4

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x + 2)$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $\frac{3}{4} \cdot \frac{4 y}{3}$ .

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Step 3.5.1.1.1

Cancel the common factor of $3$ .

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Step 3.5.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x + 2)$

Step 3.5.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 y) = \frac{3}{4} (x + 2)$

Step 3.5.1.1.2

Cancel the common factor of $4$ .

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Step 3.5.1.1.2.1

Factor $4$ out of $4 y$ .

$\frac{1}{4} (4 (y)) = \frac{3}{4} (x + 2)$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 y) = \frac{3}{4} (x + 2)$

Step 3.5.1.1.2.3

Rewrite the expression.

$y = \frac{3}{4} (x + 2)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

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

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Step 3.5.2.1.1

Apply the distributive property.

$y = \frac{3}{4} x + \frac{3}{4} \cdot 2$

Step 3.5.2.1.2

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

$y = \frac{3 x}{4} + \frac{3}{4} \cdot 2$

Step 3.5.2.1.3

Cancel the common factor of $2$ .

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Step 3.5.2.1.3.1

Factor $2$ out of $4$ .

$y = \frac{3 x}{4} + \frac{3}{2 (2)} \cdot 2$

Step 3.5.2.1.3.2

Cancel the common factor.

$y = \frac{3 x}{4} + \frac{3}{2 \cdot 2} \cdot 2$

Step 3.5.2.1.3.3

Rewrite the expression.

$y = \frac{3 x}{4} + \frac{3}{2}$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Step 5

Verify if $f^{- 1} (x) = \frac{3 x}{4} + \frac{3}{2}$ is the inverse of $f (x) = \frac{4}{3} x - 2$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\frac{4}{3} x - 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

Factor $2$ out of $\frac{4 x}{3} - 2$ .

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Step 5.2.3.1.2.1

Factor $2$ out of $\frac{4 x}{3}$ .

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

Step 5.2.3.1.2.2

Factor $2$ out of $- 2$ .

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

Step 5.2.3.1.2.3

Factor $2$ out of $2 \frac{2 x}{3} + 2 (- 1)$ .

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

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

Step 5.2.3.1.3

Multiply $3$ by $2$ .

$f^{- 1} (\frac{4}{3} x - 2) = \frac{6 (\frac{2 x}{3} - 1)}{4} + \frac{3}{2}$

Step 5.2.3.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{- 1} (\frac{4}{3} x - 2) = \frac{6 (\frac{2 x}{3} - 1 \cdot \frac{3}{3})}{4} + \frac{3}{2}$

Step 5.2.3.1.5

Combine $- 1$ and $\frac{3}{3}$ .

$f^{- 1} (\frac{4}{3} x - 2) = \frac{6 (\frac{2 x}{3} + \frac{- 1 \cdot 3}{3})}{4} + \frac{3}{2}$

Step 5.2.3.1.6

Combine the numerators over the common denominator.

$f^{- 1} (\frac{4}{3} x - 2) = \frac{6 (\frac{2 x - 1 \cdot 3}{3})}{4} + \frac{3}{2}$

Step 5.2.3.1.7

Multiply $- 1$ by $3$ .

$f^{- 1} (\frac{4}{3} x - 2) = \frac{6 (\frac{2 x - 3}{3})}{4} + \frac{3}{2}$

Step 5.2.3.2

Combine $6$ and $\frac{2 x - 3}{3}$ .

$f^{- 1} (\frac{4}{3} x - 2) = \frac{\frac{6 (2 x - 3)}{3}}{4} + \frac{3}{2}$

Step 5.2.3.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.3.1

Reduce the expression $\frac{6 (2 x - 3)}{3}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $3$ out of $6 (2 x - 3)$ .

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

Step 5.2.3.3.1.2

Factor $3$ out of $3$ .

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

Step 5.2.3.3.1.3

Cancel the common factor.

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

Step 5.2.3.3.1.4

Rewrite the expression.

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

Step 5.2.3.3.2

Divide $2 (2 x - 3)$ by $1$ .

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

Step 5.2.3.4

Cancel the common factors.

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Step 5.2.3.4.1

Factor $2$ out of $4$ .

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

Step 5.2.3.4.2

Cancel the common factor.

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

Step 5.2.3.4.3

Rewrite the expression.

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

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

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

Step 5.2.4.2

Combine the opposite terms in $2 x - 3 + 3$ .

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Step 5.2.4.2.1

Add $- 3$ and $3$ .

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

Step 5.2.4.2.2

Add $2 x$ and $0$ .

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

Step 5.2.4.3

Cancel the common factor of $2$ .

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Step 5.2.4.3.1

Cancel the common factor.

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

Step 5.2.4.3.2

Divide $x$ by $1$ .

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

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{3 x}{4} + \frac{3}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $4$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

Factor $3$ out of $3 x$ .

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

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

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Factor $2$ out of $4$ .

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

Step 5.3.3.4.2

Cancel the common factor.

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

Step 5.3.3.4.3

Rewrite the expression.

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

Step 5.3.3.5

Cancel the common factor of $3$ .

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Step 5.3.3.5.1

Cancel the common factor.

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

Step 5.3.3.5.2

Rewrite the expression.

$f (\frac{3 x}{4} + \frac{3}{2}) = x + 2 - 2$

Step 5.3.4

Combine the opposite terms in $x + 2 - 2$ .

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Step 5.3.4.1

Subtract $2$ from $2$ .

$f (\frac{3 x}{4} + \frac{3}{2}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{4} + \frac{3}{2}$ is the inverse of $f (x) = \frac{4}{3} x - 2$ .

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