Algebra Examples

$g (x) = \frac{1}{3 x} - 7$

Step 1

Write $g (x) = \frac{1}{3 x} - 7$ as an equation.

$y = \frac{1}{3 x} - 7$

Step 2

Interchange the variables.

$x = \frac{1}{3 y} - 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{3 y} - 7 = x$ .

$\frac{1}{3 y} - 7 = x$

Step 3.2

Add $7$ to both sides of the equation.

$\frac{1}{3 y} = x + 7$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 y, 1, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$3 y$

Step 3.4

Multiply each term in $\frac{1}{3 y} = x + 7$ by $3 y$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{1}{3 y} = x + 7$ by $3 y$ .

$\frac{1}{3 y} (3 y) = x (3 y) + 7 (3 y)$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Rewrite using the commutative property of multiplication.

$3 \frac{1}{3 y} y = x (3 y) + 7 (3 y)$

Step 3.4.2.2

Cancel the common factor of $3$ .

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Step 3.4.2.2.1

Factor $3$ out of $3 y$ .

$3 \frac{1}{3 (y)} y = x (3 y) + 7 (3 y)$

Step 3.4.2.2.2

Cancel the common factor.

$3 \frac{1}{3 y} y = x (3 y) + 7 (3 y)$

Step 3.4.2.2.3

Rewrite the expression.

$\frac{1}{y} y = x (3 y) + 7 (3 y)$

Step 3.4.2.3

Cancel the common factor of $y$ .

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Step 3.4.2.3.1

Cancel the common factor.

$\frac{1}{y} y = x (3 y) + 7 (3 y)$

Step 3.4.2.3.2

Rewrite the expression.

$1 = x (3 y) + 7 (3 y)$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Rewrite using the commutative property of multiplication.

$1 = 3 x y + 7 (3 y)$

Step 3.4.3.1.2

Multiply $3$ by $7$ .

$1 = 3 x y + 21 y$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $3 x y + 21 y = 1$ .

$3 x y + 21 y = 1$

Step 3.5.2

Factor $3 y$ out of $3 x y + 21 y$ .

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Step 3.5.2.1

Factor $3 y$ out of $3 x y$ .

$3 y (x) + 21 y = 1$

Step 3.5.2.2

Factor $3 y$ out of $21 y$ .

$3 y (x) + 3 y (7) = 1$

Step 3.5.2.3

Factor $3 y$ out of $3 y (x) + 3 y (7)$ .

$3 y (x + 7) = 1$

Step 3.5.3

Divide each term in $3 y (x + 7) = 1$ by $3 (x + 7)$ and simplify.

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Step 3.5.3.1

Divide each term in $3 y (x + 7) = 1$ by $3 (x + 7)$ .

$\frac{3 y (x + 7)}{3 (x + 7)} = \frac{1}{3 (x + 7)}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $3$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{3 y (x + 7)}{3 (x + 7)} = \frac{1}{3 (x + 7)}$

Step 3.5.3.2.1.2

Rewrite the expression.

$\frac{y (x + 7)}{x + 7} = \frac{1}{3 (x + 7)}$

Step 3.5.3.2.2

Cancel the common factor of $x + 7$ .

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Step 3.5.3.2.2.1

Cancel the common factor.

$\frac{y (x + 7)}{x + 7} = \frac{1}{3 (x + 7)}$

Step 3.5.3.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{3 (x + 7)}$

Step 4

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

$g^{- 1} (x) = \frac{1}{3 (x + 7)}$

Step 5

Verify if $g^{- 1} (x) = \frac{1}{3 (x + 7)}$ is the inverse of $g (x) = \frac{1}{3 x} - 7$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $g^{- 1} (\frac{1}{3 x} - 7)$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (\frac{1}{3 x} - 7) = \frac{1}{3 ((\frac{1}{3 x} - 7) + 7)}$

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Add $- 7$ and $7$ .

$g^{- 1} (\frac{1}{3 x} - 7) = \frac{1}{3 (\frac{1}{3 x} + 0)}$

Step 5.2.3.2

Add $\frac{1}{3 x}$ and $0$ .

$g^{- 1} (\frac{1}{3 x} - 7) = \frac{1}{3 (\frac{1}{3 x})}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

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

$g^{- 1} (\frac{1}{3 x} - 7) = \frac{1}{\frac{3}{3 x}}$

Step 5.2.4.2

Reduce the expression $\frac{3}{3 x}$ by cancelling the common factors.

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Step 5.2.4.2.1

Cancel the common factor.

$g^{- 1} (\frac{1}{3 x} - 7) = \frac{1}{\frac{3}{3 x}}$

Step 5.2.4.2.2

Rewrite the expression.

$g^{- 1} (\frac{1}{3 x} - 7) = \frac{1}{\frac{1}{x}}$

Step 5.2.5

Multiply the numerator by the reciprocal of the denominator.

$g^{- 1} (\frac{1}{3 x} - 7) = 1 x$

Step 5.2.6

Multiply $x$ by $1$ .

$g^{- 1} (\frac{1}{3 x} - 7) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $g (\frac{1}{3 (x + 7)})$ by substituting in the value of $g^{- 1}$ into $g$ .

$g (\frac{1}{3 (x + 7)}) = \frac{1}{3 (\frac{1}{3 (x + 7)})} - 7$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Combine $3$ and $\frac{1}{3 (x + 7)}$ .

$g (\frac{1}{3 (x + 7)}) = \frac{1}{\frac{3}{3 (x + 7)}} - 7$

Step 5.3.3.2

Reduce the expression $\frac{3}{3 (x + 7)}$ by cancelling the common factors.

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Step 5.3.3.2.1

Cancel the common factor.

$g (\frac{1}{3 (x + 7)}) = \frac{1}{\frac{3}{3 (x + 7)}} - 7$

Step 5.3.3.2.2

Rewrite the expression.

$g (\frac{1}{3 (x + 7)}) = \frac{1}{\frac{1}{x + 7}} - 7$

Step 5.3.3.3

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{1}{3 (x + 7)}) = 1 (x + 7) - 7$

Step 5.3.3.4

Multiply $x + 7$ by $1$ .

$g (\frac{1}{3 (x + 7)}) = x + 7 - 7$

Step 5.3.4

Combine the opposite terms in $x + 7 - 7$ .

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Step 5.3.4.1

Subtract $7$ from $7$ .

$g (\frac{1}{3 (x + 7)}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$g (\frac{1}{3 (x + 7)}) = x$

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = \frac{1}{3 (x + 7)}$ is the inverse of $g (x) = \frac{1}{3 x} - 7$ .

$g^{- 1} (x) = \frac{1}{3 (x + 7)}$