Finite Math Examples

$h (x) = \frac{8 x + 9}{8}$

Step 1

Write $h (x) = \frac{8 x + 9}{8}$ as an equation.

$y = \frac{8 x + 9}{8}$

Step 2

Interchange the variables.

$x = \frac{8 y + 9}{8}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{8 y + 9}{8} = x$ .

$\frac{8 y + 9}{8} = x$

Step 3.2

Multiply both sides by $8$ .

$\frac{8 y + 9}{8} \cdot 8 = x \cdot 8$

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

Cancel the common factor of $8$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{8 y + 9}{8} \cdot 8 = x \cdot 8$

Step 3.3.1.1.2

Rewrite the expression.

$8 y + 9 = x \cdot 8$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Move $8$ to the left of $x$ .

$8 y + 9 = 8 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $9$ from both sides of the equation.

$8 y = 8 x - 9$

Step 3.4.2

Divide each term in $8 y = 8 x - 9$ by $8$ and simplify.

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Step 3.4.2.1

Divide each term in $8 y = 8 x - 9$ by $8$ .

$\frac{8 y}{8} = \frac{8 x}{8} + \frac{- 9}{8}$

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.

$\frac{8 y}{8} = \frac{8 x}{8} + \frac{- 9}{8}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{8 x}{8} + \frac{- 9}{8}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Cancel the common factor of $8$ .

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Step 3.4.2.3.1.1.1

Cancel the common factor.

$y = \frac{8 x}{8} + \frac{- 9}{8}$

Step 3.4.2.3.1.1.2

Divide $x$ by $1$ .

$y = x + \frac{- 9}{8}$

Step 3.4.2.3.1.2

Move the negative in front of the fraction.

$y = x - \frac{9}{8}$

Step 4

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

$h^{- 1} (x) = x - \frac{9}{8}$

Step 5

Verify if $h^{- 1} (x) = x - \frac{9}{8}$ is the inverse of $h (x) = \frac{8 x + 9}{8}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $h^{- 1} (\frac{8 x + 9}{8})$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} (\frac{8 x + 9}{8}) = (\frac{8 x + 9}{8}) - \frac{9}{8}$

Step 5.2.3

Combine the numerators over the common denominator.

$h^{- 1} (\frac{8 x + 9}{8}) = \frac{8 x + 9 - 9}{8}$

Step 5.2.4

Combine the opposite terms in $8 x + 9 - 9$ .

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Step 5.2.4.1

Subtract $9$ from $9$ .

$h^{- 1} (\frac{8 x + 9}{8}) = \frac{8 x + 0}{8}$

Step 5.2.4.2

Add $8 x$ and $0$ .

$h^{- 1} (\frac{8 x + 9}{8}) = \frac{8 x}{8}$

Step 5.2.5

Cancel the common factor of $8$ .

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Step 5.2.5.1

Cancel the common factor.

$h^{- 1} (\frac{8 x + 9}{8}) = \frac{8 x}{8}$

Step 5.2.5.2

Divide $x$ by $1$ .

$h^{- 1} (\frac{8 x + 9}{8}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $h (x - \frac{9}{8})$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (x - \frac{9}{8}) = \frac{8 (x - \frac{9}{8}) + 9}{8}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Apply the distributive property.

$h (x - \frac{9}{8}) = \frac{8 x + 8 (- \frac{9}{8}) + 9}{8}$

Step 5.3.3.2

Cancel the common factor of $8$ .

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Step 5.3.3.2.1

Move the leading negative in $- \frac{9}{8}$ into the numerator.

$h (x - \frac{9}{8}) = \frac{8 x + 8 (\frac{- 9}{8}) + 9}{8}$

Step 5.3.3.2.2

Cancel the common factor.

$h (x - \frac{9}{8}) = \frac{8 x + 8 (\frac{- 9}{8}) + 9}{8}$

Step 5.3.3.2.3

Rewrite the expression.

$h (x - \frac{9}{8}) = \frac{8 x - 9 + 9}{8}$

Step 5.3.3.3

Add $- 9$ and $9$ .

$h (x - \frac{9}{8}) = \frac{8 x + 0}{8}$

Step 5.3.3.4

Add $8 x$ and $0$ .

$h (x - \frac{9}{8}) = \frac{8 x}{8}$

Step 5.3.4

Cancel the common factor of $8$ .

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Step 5.3.4.1

Cancel the common factor.

$h (x - \frac{9}{8}) = \frac{8 x}{8}$

Step 5.3.4.2

Divide $x$ by $1$ .

$h (x - \frac{9}{8}) = x$

Step 5.4

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = x - \frac{9}{8}$ is the inverse of $h (x) = \frac{8 x + 9}{8}$ .

$h^{- 1} (x) = x - \frac{9}{8}$