Precalculus Examples

$f (x) = \frac{8 x - 6}{15}$

Step 1

Write $f (x) = \frac{8 x - 6}{15}$ as an equation.

$y = \frac{8 x - 6}{15}$

Step 2

Interchange the variables.

$x = \frac{8 y - 6}{15}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{8 y - 6}{15} = x$ .

$\frac{8 y - 6}{15} = x$

Step 3.2

Multiply both sides by $15$ .

$\frac{8 y - 6}{15} \cdot 15 = x \cdot 15$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{8 y - 6}{15} \cdot 15$ .

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Step 3.3.1.1.1

Factor $2$ out of $8 y - 6$ .

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Step 3.3.1.1.1.1

Factor $2$ out of $8 y$ .

$\frac{2 (4 y) - 6}{15} \cdot 15 = x \cdot 15$

Step 3.3.1.1.1.2

Factor $2$ out of $- 6$ .

$\frac{2 (4 y) + 2 (- 3)}{15} \cdot 15 = x \cdot 15$

Step 3.3.1.1.1.3

Factor $2$ out of $2 (4 y) + 2 (- 3)$ .

$\frac{2 (4 y - 3)}{15} \cdot 15 = x \cdot 15$

Step 3.3.1.1.2

Cancel the common factor of $15$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{2 (4 y - 3)}{15} \cdot 15 = x \cdot 15$

Step 3.3.1.1.2.2

Rewrite the expression.

$2 (4 y - 3) = x \cdot 15$

Step 3.3.1.1.3

Apply the distributive property.

$2 (4 y) + 2 \cdot - 3 = x \cdot 15$

Step 3.3.1.1.4

Multiply.

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Step 3.3.1.1.4.1

Multiply $4$ by $2$ .

$8 y + 2 \cdot - 3 = x \cdot 15$

Step 3.3.1.1.4.2

Multiply $2$ by $- 3$ .

$8 y - 6 = x \cdot 15$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Move $15$ to the left of $x$ .

$8 y - 6 = 15 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $6$ to both sides of the equation.

$8 y = 15 x + 6$

Step 3.4.2

Divide each term in $8 y = 15 x + 6$ by $8$ and simplify.

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Step 3.4.2.1

Divide each term in $8 y = 15 x + 6$ by $8$ .

$\frac{8 y}{8} = \frac{15 x}{8} + \frac{6}{8}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $8$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{15 x}{8} + \frac{6}{8}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{15 x}{8} + \frac{6}{8}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Cancel the common factor of $6$ and $8$ .

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Step 3.4.2.3.1.1

Factor $2$ out of $6$ .

$y = \frac{15 x}{8} + \frac{2 (3)}{8}$

Step 3.4.2.3.1.2

Cancel the common factors.

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Step 3.4.2.3.1.2.1

Factor $2$ out of $8$ .

$y = \frac{15 x}{8} + \frac{2 \cdot 3}{2 \cdot 4}$

Step 3.4.2.3.1.2.2

Cancel the common factor.

$y = \frac{15 x}{8} + \frac{2 \cdot 3}{2 \cdot 4}$

Step 3.4.2.3.1.2.3

Rewrite the expression.

$y = \frac{15 x}{8} + \frac{3}{4}$

Step 4

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

$f^{- 1} (x) = \frac{15 x}{8} + \frac{3}{4}$

Step 5

Verify if $f^{- 1} (x) = \frac{15 x}{8} + \frac{3}{4}$ is the inverse of $f (x) = \frac{8 x - 6}{15}$ .

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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 $f^{- 1} (\frac{8 x - 6}{15})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{15 (\frac{8 x - 6}{15})}{8} + \frac{3}{4}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Factor $2$ out of $8 x - 6$ .

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Step 5.2.3.1.1

Factor $2$ out of $8 x$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{15 (\frac{2 (4 x) - 6}{15})}{8} + \frac{3}{4}$

Step 5.2.3.1.2

Factor $2$ out of $- 6$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{15 (\frac{2 (4 x) + 2 (- 3)}{15})}{8} + \frac{3}{4}$

Step 5.2.3.1.3

Factor $2$ out of $2 (4 x) + 2 (- 3)$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{15 (\frac{2 (4 x - 3)}{15})}{8} + \frac{3}{4}$

Step 5.2.3.2

Combine $15$ and $\frac{2 (4 x - 3)}{15}$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{\frac{15 (2 (4 x - 3))}{15}}{8} + \frac{3}{4}$

Step 5.2.3.3

Multiply $15$ by $2$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{\frac{30 (4 x - 3)}{15}}{8} + \frac{3}{4}$

Step 5.2.3.4

Reduce the expression by cancelling the common factors.

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Step 5.2.3.4.1

Reduce the expression $\frac{30 (4 x - 3)}{15}$ by cancelling the common factors.

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Step 5.2.3.4.1.1

Factor $15$ out of $30 (4 x - 3)$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{\frac{15 (2 (4 x - 3))}{15}}{8} + \frac{3}{4}$

Step 5.2.3.4.1.2

Factor $15$ out of $15$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{\frac{15 (2 (4 x - 3))}{15 (1)}}{8} + \frac{3}{4}$

Step 5.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{\frac{15 (2 (4 x - 3))}{15 \cdot 1}}{8} + \frac{3}{4}$

Step 5.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{\frac{2 (4 x - 3)}{1}}{8} + \frac{3}{4}$

Step 5.2.3.4.2

Divide $2 (4 x - 3)$ by $1$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{2 (4 x - 3)}{8} + \frac{3}{4}$

Step 5.2.3.5

Cancel the common factors.

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Step 5.2.3.5.1

Factor $2$ out of $8$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{2 (4 x - 3)}{2 \cdot 4} + \frac{3}{4}$

Step 5.2.3.5.2

Cancel the common factor.

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{2 (4 x - 3)}{2 \cdot 4} + \frac{3}{4}$

Step 5.2.3.5.3

Rewrite the expression.

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{4 x - 3}{4} + \frac{3}{4}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{4 x - 3 + 3}{4}$

Step 5.2.4.2

Combine the opposite terms in $4 x - 3 + 3$ .

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Step 5.2.4.2.1

Add $- 3$ and $3$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{4 x + 0}{4}$

Step 5.2.4.2.2

Add $4 x$ and $0$ .

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{4 x}{4}$

Step 5.2.4.3

Cancel the common factor of $4$ .

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Step 5.2.4.3.1

Cancel the common factor.

$f^{- 1} (\frac{8 x - 6}{15}) = \frac{4 x}{4}$

Step 5.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{8 x - 6}{15}) = x$

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 $f (\frac{15 x}{8} + \frac{3}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{8 (\frac{15 x}{8} + \frac{3}{4}) - 6}{15}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $2$ out of $8 (\frac{15 x}{8} + \frac{3}{4}) - 6$ .

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Step 5.3.3.1.1

Factor $2$ out of $8 (\frac{15 x}{8} + \frac{3}{4})$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (4 (\frac{15 x}{8} + \frac{3}{4})) - 6}{15}$

Step 5.3.3.1.2

Factor $2$ out of $- 6$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (4 (\frac{15 x}{8} + \frac{3}{4})) + 2 (- 3)}{15}$

Step 5.3.3.1.3

Factor $2$ out of $2 (4 (\frac{15 x}{8} + \frac{3}{4})) + 2 (- 3)$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (4 (\frac{15 x}{8} + \frac{3}{4}) - 3)}{15}$

Step 5.3.3.2

Apply the distributive property.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (4 (\frac{15 x}{8}) + 4 (\frac{3}{4}) - 3)}{15}$

Step 5.3.3.3

Cancel the common factor of $4$ .

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Step 5.3.3.3.1

Factor $4$ out of $8$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (4 (\frac{15 x}{4 (2)}) + 4 (\frac{3}{4}) - 3)}{15}$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (4 (\frac{15 x}{4 \cdot 2}) + 4 (\frac{3}{4}) - 3)}{15}$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (\frac{15 x}{2} + 4 (\frac{3}{4}) - 3)}{15}$

Step 5.3.3.4

Cancel the common factor of $4$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (\frac{15 x}{2} + 4 (\frac{3}{4}) - 3)}{15}$

Step 5.3.3.4.2

Rewrite the expression.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (\frac{15 x}{2} + 3 - 3)}{15}$

Step 5.3.3.5

Subtract $3$ from $3$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (\frac{15 x}{2} + 0)}{15}$

Step 5.3.3.6

Add $\frac{15 x}{2}$ and $0$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{2 (\frac{15 x}{2})}{15}$

Step 5.3.4

Combine $2$ and $\frac{15 x}{2}$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{\frac{2 (15 x)}{2}}{15}$

Step 5.3.5

Multiply $2$ by $15$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{\frac{30 x}{2}}{15}$

Step 5.3.6

Reduce the expression by cancelling the common factors.

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Step 5.3.6.1

Reduce the expression $\frac{30 x}{2}$ by cancelling the common factors.

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Step 5.3.6.1.1

Factor $2$ out of $30 x$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{\frac{2 (15 x)}{2}}{15}$

Step 5.3.6.1.2

Factor $2$ out of $2$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{\frac{2 (15 x)}{2 (1)}}{15}$

Step 5.3.6.1.3

Cancel the common factor.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{\frac{2 (15 x)}{2 \cdot 1}}{15}$

Step 5.3.6.1.4

Rewrite the expression.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{\frac{15 x}{1}}{15}$

Step 5.3.6.2

Divide $15 x$ by $1$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{15 x}{15}$

Step 5.3.7

Cancel the common factor of $15$ .

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Step 5.3.7.1

Cancel the common factor.

$f (\frac{15 x}{8} + \frac{3}{4}) = \frac{15 x}{15}$

Step 5.3.7.2

Divide $x$ by $1$ .

$f (\frac{15 x}{8} + \frac{3}{4}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{15 x}{8} + \frac{3}{4}$ is the inverse of $f (x) = \frac{8 x - 6}{15}$ .

$f^{- 1} (x) = \frac{15 x}{8} + \frac{3}{4}$