Algebra Examples

$8 (x - 3) + 4$

Step 1

Interchange the variables.

$x = 8 (y - 3) + 4$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $8 (y - 3) + 4 = x$ .

$8 (y - 3) + 4 = x$

Step 2.2

Simplify $8 (y - 3) + 4$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$8 y + 8 \cdot - 3 + 4 = x$

Step 2.2.1.2

Multiply $8$ by $- 3$ .

$8 y - 24 + 4 = x$

Step 2.2.2

Add $- 24$ and $4$ .

$8 y - 20 = x$

Step 2.3

Add $20$ to both sides of the equation.

$8 y = x + 20$

Step 2.4

Divide each term in $8 y = x + 20$ by $8$ and simplify.

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Step 2.4.1

Divide each term in $8 y = x + 20$ by $8$ .

$\frac{8 y}{8} = \frac{x}{8} + \frac{20}{8}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $8$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{x}{8} + \frac{20}{8}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{8} + \frac{20}{8}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Cancel the common factor of $20$ and $8$ .

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Step 2.4.3.1.1

Factor $4$ out of $20$ .

$y = \frac{x}{8} + \frac{4 (5)}{8}$

Step 2.4.3.1.2

Cancel the common factors.

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Step 2.4.3.1.2.1

Factor $4$ out of $8$ .

$y = \frac{x}{8} + \frac{4 \cdot 5}{4 \cdot 2}$

Step 2.4.3.1.2.2

Cancel the common factor.

$y = \frac{x}{8} + \frac{4 \cdot 5}{4 \cdot 2}$

Step 2.4.3.1.2.3

Rewrite the expression.

$y = \frac{x}{8} + \frac{5}{2}$

Step 3

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

$f^{- 1} (x) = \frac{x}{8} + \frac{5}{2}$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{8} + \frac{5}{2}$ is the inverse of $f (x) = 8 (x - 3) + 4$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

$f^{- 1} (8 (x - 3) + 4) = \frac{8 (x - 3) + 4}{8} + \frac{5}{2}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Cancel the common factor of $8 (x - 3) + 4$ and $8$ .

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Step 4.2.3.1.1

Factor $4$ out of $8 (x - 3)$ .

$f^{- 1} (8 (x - 3) + 4) = \frac{4 (2 (x - 3)) + 4}{8} + \frac{5}{2}$

Step 4.2.3.1.2

Factor $4$ out of $4$ .

$f^{- 1} (8 (x - 3) + 4) = \frac{4 (2 (x - 3)) + 4 (1)}{8} + \frac{5}{2}$

Step 4.2.3.1.3

Factor $4$ out of $4 (2 (x - 3)) + 4 (1)$ .

$f^{- 1} (8 (x - 3) + 4) = \frac{4 (2 (x - 3) + 1)}{8} + \frac{5}{2}$

Step 4.2.3.1.4

Cancel the common factors.

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Step 4.2.3.1.4.1

Factor $4$ out of $8$ .

$f^{- 1} (8 (x - 3) + 4) = \frac{4 (2 (x - 3) + 1)}{4 (2)} + \frac{5}{2}$

Step 4.2.3.1.4.2

Cancel the common factor.

$f^{- 1} (8 (x - 3) + 4) = \frac{4 (2 (x - 3) + 1)}{4 \cdot 2} + \frac{5}{2}$

Step 4.2.3.1.4.3

Rewrite the expression.

$f^{- 1} (8 (x - 3) + 4) = \frac{2 (x - 3) + 1}{2} + \frac{5}{2}$

Step 4.2.3.2

Simplify the numerator.

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Step 4.2.3.2.1

Apply the distributive property.

$f^{- 1} (8 (x - 3) + 4) = \frac{2 x + 2 \cdot - 3 + 1}{2} + \frac{5}{2}$

Step 4.2.3.2.2

Multiply $2$ by $- 3$ .

$f^{- 1} (8 (x - 3) + 4) = \frac{2 x - 6 + 1}{2} + \frac{5}{2}$

Step 4.2.3.2.3

Add $- 6$ and $1$ .

$f^{- 1} (8 (x - 3) + 4) = \frac{2 x - 5}{2} + \frac{5}{2}$

Step 4.2.4

Simplify terms.

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Step 4.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (8 (x - 3) + 4) = \frac{2 x - 5 + 5}{2}$

Step 4.2.4.2

Combine the opposite terms in $2 x - 5 + 5$ .

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Step 4.2.4.2.1

Add $- 5$ and $5$ .

$f^{- 1} (8 (x - 3) + 4) = \frac{2 x + 0}{2}$

Step 4.2.4.2.2

Add $2 x$ and $0$ .

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

Step 4.2.4.3

Cancel the common factor of $2$ .

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Step 4.2.4.3.1

Cancel the common factor.

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

Step 4.2.4.3.2

Divide $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x}{8} + \frac{5}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{8} + \frac{5}{2}) = 8 ((\frac{x}{8} + \frac{5}{2}) - 3) + 4$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8} + \frac{5}{2} - 3 \cdot \frac{2}{2}) + 4$

Step 4.3.3.2

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

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8} + \frac{5}{2} + \frac{- 3 \cdot 2}{2}) + 4$

Step 4.3.3.3

Combine the numerators over the common denominator.

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8} + \frac{5 - 3 \cdot 2}{2}) + 4$

Step 4.3.3.4

Simplify the numerator.

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Step 4.3.3.4.1

Multiply $- 3$ by $2$ .

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8} + \frac{5 - 6}{2}) + 4$

Step 4.3.3.4.2

Subtract $6$ from $5$ .

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8} + \frac{- 1}{2}) + 4$

Step 4.3.3.5

Move the negative in front of the fraction.

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8} - \frac{1}{2}) + 4$

Step 4.3.3.6

Apply the distributive property.

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8}) + 8 (- \frac{1}{2}) + 4$

Step 4.3.3.7

Cancel the common factor of $8$ .

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Step 4.3.3.7.1

Cancel the common factor.

$f (\frac{x}{8} + \frac{5}{2}) = 8 (\frac{x}{8}) + 8 (- \frac{1}{2}) + 4$

Step 4.3.3.7.2

Rewrite the expression.

$f (\frac{x}{8} + \frac{5}{2}) = x + 8 (- \frac{1}{2}) + 4$

Step 4.3.3.8

Cancel the common factor of $2$ .

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Step 4.3.3.8.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\frac{x}{8} + \frac{5}{2}) = x + 8 (\frac{- 1}{2}) + 4$

Step 4.3.3.8.2

Factor $2$ out of $8$ .

$f (\frac{x}{8} + \frac{5}{2}) = x + 2 (4) (\frac{- 1}{2}) + 4$

Step 4.3.3.8.3

Cancel the common factor.

$f (\frac{x}{8} + \frac{5}{2}) = x + 2 \cdot (4 (\frac{- 1}{2})) + 4$

Step 4.3.3.8.4

Rewrite the expression.

$f (\frac{x}{8} + \frac{5}{2}) = x + 4 \cdot - 1 + 4$

Step 4.3.3.9

Multiply $4$ by $- 1$ .

$f (\frac{x}{8} + \frac{5}{2}) = x - 4 + 4$

Step 4.3.4

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

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Step 4.3.4.1

Add $- 4$ and $4$ .

$f (\frac{x}{8} + \frac{5}{2}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{x}{8} + \frac{5}{2}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{8} + \frac{5}{2}$ is the inverse of $f (x) = 8 (x - 3) + 4$ .

$f^{- 1} (x) = \frac{x}{8} + \frac{5}{2}$