Algebra Examples

$h (x) = - \frac{9 x}{4} - \frac{1}{3}$

Step 1

Write $h (x) = - \frac{9 x}{4} - \frac{1}{3}$ as an equation.

$y = - \frac{9 x}{4} - \frac{1}{3}$

Step 2

Interchange the variables.

$x = - \frac{9 y}{4} - \frac{1}{3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{9 y}{4} - \frac{1}{3} = x$ .

$- \frac{9 y}{4} - \frac{1}{3} = x$

Step 3.2

Add $\frac{1}{3}$ to both sides of the equation.

$- \frac{9 y}{4} = x + \frac{1}{3}$

Step 3.3

Multiply both sides of the equation by $- \frac{4}{9}$ .

$- \frac{4}{9} (- \frac{9 y}{4}) = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $- \frac{4}{9} (- \frac{9 y}{4})$ .

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Step 3.4.1.1.1

Cancel the common factor of $4$ .

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Step 3.4.1.1.1.1

Move the leading negative in $- \frac{4}{9}$ into the numerator.

$\frac{- 4}{9} (- \frac{9 y}{4}) = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.1.2

Move the leading negative in $- \frac{9 y}{4}$ into the numerator.

$\frac{- 4}{9} \cdot \frac{- 9 y}{4} = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.1.3

Factor $4$ out of $- 4$ .

$\frac{4 (- 1)}{9} \cdot \frac{- 9 y}{4} = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.1.4

Cancel the common factor.

$\frac{4 \cdot - 1}{9} \cdot \frac{- 9 y}{4} = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{9} (- 9 y) = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.2

Cancel the common factor of $9$ .

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Step 3.4.1.1.2.1

Factor $9$ out of $- 9 y$ .

$\frac{- 1}{9} (9 (- y)) = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{9} (9 (- y)) = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.2.3

Rewrite the expression.

$- - y = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.3

Multiply.

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Step 3.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{4}{9} (x + \frac{1}{3})$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \frac{4}{9} (x + \frac{1}{3})$ .

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Step 3.4.2.1.1

Apply the distributive property.

$y = - \frac{4}{9} x - \frac{4}{9} \cdot \frac{1}{3}$

Step 3.4.2.1.2

Combine $x$ and $\frac{4}{9}$ .

$y = - \frac{x \cdot 4}{9} - \frac{4}{9} \cdot \frac{1}{3}$

Step 3.4.2.1.3

Multiply $- \frac{4}{9} \cdot \frac{1}{3}$ .

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Step 3.4.2.1.3.1

Multiply $\frac{1}{3}$ by $\frac{4}{9}$ .

$y = - \frac{x \cdot 4}{9} - \frac{4}{3 \cdot 9}$

Step 3.4.2.1.3.2

Multiply $3$ by $9$ .

$y = - \frac{x \cdot 4}{9} - \frac{4}{27}$

Step 3.4.2.1.4

Move $4$ to the left of $x$ .

$y = - \frac{4 x}{9} - \frac{4}{27}$

Step 4

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

$h^{- 1} (x) = - \frac{4 x}{9} - \frac{4}{27}$

Step 5

Verify if $h^{- 1} (x) = - \frac{4 x}{9} - \frac{4}{27}$ is the inverse of $h (x) = - \frac{9 x}{4} - \frac{1}{3}$ .

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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 $h^{- 1} (- \frac{9 x}{4} - \frac{1}{3})$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{4 (- \frac{9 x}{4} - \frac{1}{3})}{9} - \frac{4}{27}$

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

Factor $- 1$ out of $- \frac{9 x}{4} - \frac{1}{3}$ .

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Step 5.2.3.1.1.1

Factor $- 1$ out of $- \frac{9 x}{4}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{4 (- (\frac{9 x}{4}) - \frac{1}{3})}{9} - \frac{4}{27}$

Step 5.2.3.1.1.2

Factor $- 1$ out of $- \frac{1}{3}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{4 (- (\frac{9 x}{4}) - (\frac{1}{3}))}{9} - \frac{4}{27}$

Step 5.2.3.1.1.3

Factor $- 1$ out of $- (\frac{9 x}{4}) - (\frac{1}{3})$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{4 (- (\frac{9 x}{4} + \frac{1}{3}))}{9} - \frac{4}{27}$

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{4 \cdot (- 1 (\frac{9 x}{4} + \frac{1}{3}))}{9} - \frac{4}{27}$

Step 5.2.3.1.2

Combine exponents.

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Step 5.2.3.1.2.1

Factor out negative.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- (4 (\frac{9 x}{4} + \frac{1}{3}))}{9} - \frac{4}{27}$

Step 5.2.3.1.2.2

Multiply $4$ by $- 1$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 (\frac{9 x}{4} + \frac{1}{3})}{9} - \frac{4}{27}$

Step 5.2.3.1.3

To write $\frac{9 x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 (\frac{9 x}{4} \cdot \frac{3}{3} + \frac{1}{3})}{9} - \frac{4}{27}$

Step 5.2.3.1.4

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 (\frac{9 x}{4} \cdot \frac{3}{3} + \frac{1}{3} \cdot \frac{4}{4})}{9} - \frac{4}{27}$

Step 5.2.3.1.5

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.3.1.5.1

Multiply $\frac{9 x}{4}$ by $\frac{3}{3}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 (\frac{9 x \cdot 3}{4 \cdot 3} + \frac{1}{3} \cdot \frac{4}{4})}{9} - \frac{4}{27}$

Step 5.2.3.1.5.2

Multiply $4$ by $3$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 (\frac{9 x \cdot 3}{12} + \frac{1}{3} \cdot \frac{4}{4})}{9} - \frac{4}{27}$

Step 5.2.3.1.5.3

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 (\frac{9 x \cdot 3}{12} + \frac{4}{3 \cdot 4})}{9} - \frac{4}{27}$

Step 5.2.3.1.5.4

Multiply $3$ by $4$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 (\frac{9 x \cdot 3}{12} + \frac{4}{12})}{9} - \frac{4}{27}$

Step 5.2.3.1.6

Combine the numerators over the common denominator.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 \frac{9 x \cdot 3 + 4}{12}}{9} - \frac{4}{27}$

Step 5.2.3.1.7

Multiply $3$ by $9$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- 4 \frac{27 x + 4}{12}}{9} - \frac{4}{27}$

Step 5.2.3.2

Combine $- 4$ and $\frac{27 x + 4}{12}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{\frac{- 4 (27 x + 4)}{12}}{9} - \frac{4}{27}$

Step 5.2.3.3

Simplify the numerator.

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Step 5.2.3.3.1

Reduce the expression $\frac{- 4 (27 x + 4)}{12}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $4$ out of $- 4 (27 x + 4)$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{\frac{4 (- (27 x + 4))}{12}}{9} - \frac{4}{27}$

Step 5.2.3.3.1.2

Factor $4$ out of $12$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{\frac{4 (- (27 x + 4))}{4 (3)}}{9} - \frac{4}{27}$

Step 5.2.3.3.1.3

Cancel the common factor.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{\frac{4 (- (27 x + 4))}{4 \cdot 3}}{9} - \frac{4}{27}$

Step 5.2.3.3.1.4

Rewrite the expression.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{\frac{- (27 x + 4)}{3}}{9} - \frac{4}{27}$

Step 5.2.3.3.2

Move the negative in front of the fraction.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - \frac{- \frac{27 x + 4}{3}}{9} - \frac{4}{27}$

Step 5.2.3.4

Multiply the numerator by the reciprocal of the denominator.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = - (- \frac{27 x + 4}{3} \cdot \frac{1}{9}) - \frac{4}{27}$

Step 5.2.3.5

Multiply $- \frac{27 x + 4}{3} \cdot \frac{1}{9}$ .

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Step 5.2.3.5.1

Multiply $\frac{1}{9}$ by $\frac{27 x + 4}{3}$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = \frac{27 x + 4}{9 \cdot 3} - \frac{4}{27}$

Step 5.2.3.5.2

Multiply $9$ by $3$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = \frac{27 x + 4}{27} - \frac{4}{27}$

Step 5.2.3.6

Multiply $- - \frac{27 x + 4}{27}$ .

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Step 5.2.3.6.1

Multiply $- 1$ by $- 1$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = 1 (\frac{27 x + 4}{27}) - \frac{4}{27}$

Step 5.2.3.6.2

Multiply $\frac{27 x + 4}{27}$ by $1$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = \frac{27 x + 4}{27} - \frac{4}{27}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = \frac{27 x + 4 - 4}{27}$

Step 5.2.4.2

Combine the opposite terms in $27 x + 4 - 4$ .

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Step 5.2.4.2.1

Subtract $4$ from $4$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = \frac{27 x + 0}{27}$

Step 5.2.4.2.2

Add $27 x$ and $0$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = \frac{27 x}{27}$

Step 5.2.4.3

Cancel the common factor of $27$ .

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Step 5.2.4.3.1

Cancel the common factor.

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = \frac{27 x}{27}$

Step 5.2.4.3.2

Divide $x$ by $1$ .

$h^{- 1} (- \frac{9 x}{4} - \frac{1}{3}) = x$

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 $h (- \frac{4 x}{9} - \frac{4}{27})$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{9 (- \frac{4 x}{9} - \frac{4}{27})}{4} - \frac{1}{3}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

Factor $- 1$ out of $- \frac{4 x}{9} - \frac{4}{27}$ .

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Step 5.3.3.1.1.1

Factor $- 1$ out of $- \frac{4 x}{9}$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{9 (- (\frac{4 x}{9}) - \frac{4}{27})}{4} - \frac{1}{3}$

Step 5.3.3.1.1.2

Factor $- 1$ out of $- \frac{4}{27}$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{9 (- (\frac{4 x}{9}) - (\frac{4}{27}))}{4} - \frac{1}{3}$

Step 5.3.3.1.1.3

Factor $- 1$ out of $- (\frac{4 x}{9}) - (\frac{4}{27})$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{9 (- (\frac{4 x}{9} + \frac{4}{27}))}{4} - \frac{1}{3}$

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{9 \cdot (- 1 (\frac{4 x}{9} + \frac{4}{27}))}{4} - \frac{1}{3}$

Step 5.3.3.1.2

Combine exponents.

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Step 5.3.3.1.2.1

Factor out negative.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- (9 (\frac{4 x}{9} + \frac{4}{27}))}{4} - \frac{1}{3}$

Step 5.3.3.1.2.2

Multiply $9$ by $- 1$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 (\frac{4 x}{9} + \frac{4}{27})}{4} - \frac{1}{3}$

Step 5.3.3.1.3

To write $\frac{4 x}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 (\frac{4 x}{9} \cdot \frac{3}{3} + \frac{4}{27})}{4} - \frac{1}{3}$

Step 5.3.3.1.4

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.1.4.1

Multiply $\frac{4 x}{9}$ by $\frac{3}{3}$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 (\frac{4 x \cdot 3}{9 \cdot 3} + \frac{4}{27})}{4} - \frac{1}{3}$

Step 5.3.3.1.4.2

Multiply $9$ by $3$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 (\frac{4 x \cdot 3}{27} + \frac{4}{27})}{4} - \frac{1}{3}$

Step 5.3.3.1.5

Combine the numerators over the common denominator.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 \frac{4 x \cdot 3 + 4}{27}}{4} - \frac{1}{3}$

Step 5.3.3.1.6

Simplify the numerator.

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Step 5.3.3.1.6.1

Factor $4$ out of $4 x \cdot 3 + 4$ .

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Step 5.3.3.1.6.1.1

Factor $4$ out of $4 x \cdot 3$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 \frac{4 (x \cdot 3) + 4}{27}}{4} - \frac{1}{3}$

Step 5.3.3.1.6.1.2

Factor $4$ out of $4$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 \frac{4 (x \cdot 3) + 4 (1)}{27}}{4} - \frac{1}{3}$

Step 5.3.3.1.6.1.3

Factor $4$ out of $4 (x \cdot 3) + 4 (1)$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 \frac{4 (x \cdot 3 + 1)}{27}}{4} - \frac{1}{3}$

Step 5.3.3.1.6.2

Move $3$ to the left of $x$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- 9 \frac{4 (3 x + 1)}{27}}{4} - \frac{1}{3}$

Step 5.3.3.2

Combine $- 9$ and $\frac{4 (3 x + 1)}{27}$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{\frac{- 9 (4 (3 x + 1))}{27}}{4} - \frac{1}{3}$

Step 5.3.3.3

Multiply $- 9$ by $4$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{\frac{- 36 (3 x + 1)}{27}}{4} - \frac{1}{3}$

Step 5.3.3.4

Simplify the numerator.

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Step 5.3.3.4.1

Reduce the expression $\frac{- 36 (3 x + 1)}{27}$ by cancelling the common factors.

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Step 5.3.3.4.1.1

Factor $9$ out of $- 36 (3 x + 1)$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{\frac{9 (- 4 (3 x + 1))}{27}}{4} - \frac{1}{3}$

Step 5.3.3.4.1.2

Factor $9$ out of $27$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{\frac{9 (- 4 (3 x + 1))}{9 (3)}}{4} - \frac{1}{3}$

Step 5.3.3.4.1.3

Cancel the common factor.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{\frac{9 (- 4 (3 x + 1))}{9 \cdot 3}}{4} - \frac{1}{3}$

Step 5.3.3.4.1.4

Rewrite the expression.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{\frac{- 4 (3 x + 1)}{3}}{4} - \frac{1}{3}$

Step 5.3.3.4.2

Move the negative in front of the fraction.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- \frac{4 (3 x + 1)}{3}}{4} - \frac{1}{3}$

Step 5.3.3.5

Multiply the numerator by the reciprocal of the denominator.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - (- \frac{4 (3 x + 1)}{3} \cdot \frac{1}{4}) - \frac{1}{3}$

Step 5.3.3.6

Cancel the common factor of $4$ .

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Step 5.3.3.6.1

Move the leading negative in $- \frac{4 (3 x + 1)}{3}$ into the numerator.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - (\frac{- 4 (3 x + 1)}{3} \cdot \frac{1}{4}) - \frac{1}{3}$

Step 5.3.3.6.2

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

$h (- \frac{4 x}{9} - \frac{4}{27}) = - (\frac{4 (- (3 x + 1))}{3} \cdot \frac{1}{4}) - \frac{1}{3}$

Step 5.3.3.6.3

Cancel the common factor.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - (\frac{4 (- (3 x + 1))}{3} \cdot \frac{1}{4}) - \frac{1}{3}$

Step 5.3.3.6.4

Rewrite the expression.

$h (- \frac{4 x}{9} - \frac{4}{27}) = - \frac{- (3 x + 1)}{3} - \frac{1}{3}$

Step 5.3.3.7

Move the negative in front of the fraction.

$h (- \frac{4 x}{9} - \frac{4}{27}) = \frac{3 x + 1}{3} - \frac{1}{3}$

Step 5.3.3.8

Multiply $- - \frac{3 x + 1}{3}$ .

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Step 5.3.3.8.1

Multiply $- 1$ by $- 1$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = 1 (\frac{3 x + 1}{3}) - \frac{1}{3}$

Step 5.3.3.8.2

Multiply $\frac{3 x + 1}{3}$ by $1$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = \frac{3 x + 1}{3} - \frac{1}{3}$

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine the numerators over the common denominator.

$h (- \frac{4 x}{9} - \frac{4}{27}) = \frac{3 x + 1 - 1}{3}$

Step 5.3.4.2

Combine the opposite terms in $3 x + 1 - 1$ .

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Step 5.3.4.2.1

Subtract $1$ from $1$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = \frac{3 x + 0}{3}$

Step 5.3.4.2.2

Add $3 x$ and $0$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = \frac{3 x}{3}$

Step 5.3.4.3

Cancel the common factor of $3$ .

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Step 5.3.4.3.1

Cancel the common factor.

$h (- \frac{4 x}{9} - \frac{4}{27}) = \frac{3 x}{3}$

Step 5.3.4.3.2

Divide $x$ by $1$ .

$h (- \frac{4 x}{9} - \frac{4}{27}) = x$

Step 5.4

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = - \frac{4 x}{9} - \frac{4}{27}$ is the inverse of $h (x) = - \frac{9 x}{4} - \frac{1}{3}$ .

$h^{- 1} (x) = - \frac{4 x}{9} - \frac{4}{27}$