Find the Inverse f(x)=(3x+4)/(x-1)

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Multiply the equation by $y - 1$ .

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

Simplify the left side.

Simplify $(y - 1) (x)$ .

Apply the distributive property.

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

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

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

Simplify the right side.

Cancel the common factor of $y - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$y x - x = 3 y + 4$

Solve for $y$ .

Subtract $3 y$ from both sides of the equation.

$y x - x - 3 y = 4$

Add $x$ to both sides of the equation.

$y x - 3 y = 4 + x$

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

Factor $y$ out of $y x$ .

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

Factor $y$ out of $- 3 y$ .

$y (x) + y \cdot - 3 = 4 + x$

Factor $y$ out of $y (x) + y \cdot - 3$ .

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

Divide each term in $y (x - 3) = 4 + x$ by $x - 3$ and simplify.

Divide each term in $y (x - 3) = 4 + x$ by $x - 3$ .

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

Simplify the left side.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

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

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

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

Remove parentheses.

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

Multiply the numerator and denominator of the complex fraction by $x - 1$ .

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Apply the distributive property.

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

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

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

Multiply $- 1$ by $4$ .

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

Add $4 x$ and $3 x$ .

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x - 4 + 4}{3 x + 4 + (x - 1) \cdot - 3}$

Add $- 4$ and $4$ .

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x + 0}{3 x + 4 + (x - 1) \cdot - 3}$

Add $7 x$ and $0$ .

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{3 x + 4 + (x - 1) \cdot - 3}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{3 x + 4 + x \cdot - 3 - 1 \cdot - 3}$

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

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{3 x + 4 - 3 \cdot x - 1 \cdot - 3}$

Multiply $- 1$ by $- 3$ .

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{3 x + 4 - 3 \cdot x + 3}$

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{0 + 4 + 3}$

Add $0$ and $4$ .

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{4 + 3}$

Add $4$ and $3$ .

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x + 4}{x - 1}) = \frac{7 x}{7}$

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Multiply the numerator and denominator of the complex fraction by $x - 3$ .

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

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the numerator.

Factor $x - 3$ out of $(x - 3) \cdot 3 \frac{4 + x}{x - 3} + (x - 3) \cdot 4$ .

Factor $x - 3$ out of $(x - 3) \cdot 3 \frac{4 + x}{x - 3}$ .

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

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

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

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

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

Combine $3$ and $\frac{4 + x}{x - 3}$ .

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

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

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

Combine the numerators over the common denominator.

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

Reorder terms.

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

Rewrite $\frac{4 (x - 3) + 3 (4 + x)}{x - 3}$ in a factored form.

Apply the distributive property.

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

Multiply $4$ by $- 3$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{4 x - 12 + 3 (4 + x)}{x - 3})}{4 + x + (x - 3) \cdot - 1}$

Apply the distributive property.

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{4 x - 12 + 3 \cdot 4 + 3 x}{x - 3})}{4 + x + (x - 3) \cdot - 1}$

Multiply $3$ by $4$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{4 x - 12 + 12 + 3 x}{x - 3})}{4 + x + (x - 3) \cdot - 1}$

Add $4 x$ and $3 x$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x - 12 + 12}{x - 3})}{4 + x + (x - 3) \cdot - 1}$

Add $- 12$ and $12$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x + 0}{x - 3})}{4 + x + (x - 3) \cdot - 1}$

Add $7 x$ and $0$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{4 + x + (x - 3) \cdot - 1}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{4 + x + x \cdot - 1 - 3 \cdot - 1}$

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

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{4 + x - 1 \cdot x - 3 \cdot - 1}$

Multiply $- 3$ by $- 1$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{4 + x - 1 \cdot x + 3}$

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

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{4 + x - x + 3}$

Add $4$ and $3$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{x - x + 7}$

Subtract $x$ from $x$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{0 + 7}$

Add $0$ and $7$ .

$f (\frac{4 + x}{x - 3}) = \frac{(x - 3) (\frac{7 x}{x - 3})}{7}$

Reduce the expression by cancelling the common factors.

Factor $\frac{7 x}{x - 3}$ out of $\frac{(x - 3) \frac{7 x}{x - 3}}{7}$ .

$f (\frac{4 + x}{x - 3}) = \frac{7 x}{x - 3} \cdot \frac{x - 3}{7}$

Cancel the common factor of $7$ .

Factor $7$ out of $7 x$ .

$f (\frac{4 + x}{x - 3}) = \frac{7 (x)}{x - 3} \cdot \frac{x - 3}{7}$

Cancel the common factor.

$f (\frac{4 + x}{x - 3}) = \frac{7 x}{x - 3} \cdot \frac{x - 3}{7}$

Rewrite the expression.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Find the Inverse f(x)=(3x+4)/(x-1)

Related Questions

Support LCMGCF.com by disabling your adblocker.