Find the Inverse 7/(3x-1)

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

Find the LCD of the terms in the equation.

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

$3 y - 1, 1$

Remove parentheses.

$3 y - 1, 1$

The LCM of one and any expression is the expression.

$3 y - 1$

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

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

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

Simplify the left side.

Cancel the common factor of $3 y - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$7 = x (3 y - 1)$

Simplify the right side.

Simplify by multiplying through.

Apply the distributive property.

$7 = x (3 y) + x \cdot - 1$

Reorder.

Rewrite using the commutative property of multiplication.

$7 = 3 x y + x \cdot - 1$

Move $- 1$ to the left of $x$ .

$7 = 3 x y - 1 \cdot x$

Rewrite $- 1 x$ as $- x$ .

$7 = 3 x y - x$

Solve the equation.

Rewrite the equation as $3 x y - x = 7$ .

$3 x y - x = 7$

Add $x$ to both sides of the equation.

$3 x y = 7 + x$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Combine $3$ and $\frac{7}{3 x - 1}$ .

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

Multiply $3$ by $7$ .

$f^{- 1} (\frac{7}{3 x - 1}) = \frac{7}{\frac{21}{3 x - 1}} + \frac{1}{3}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{7}{3 x - 1}) = 7 (\frac{3 x - 1}{21}) + \frac{1}{3}$

Cancel the common factor of $7$ .

Factor $7$ out of $21$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify terms.

Combine the numerators over the common denominator.

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

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

Add $- 1$ and $1$ .

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

Add $3 x$ and $0$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Simplify the denominator.

Apply the distributive property.

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $1$ from $1$ .

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

Add $\frac{7}{x}$ and $0$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Find the Inverse 7/(3x-1)

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