Find the Inverse (7x+3)/(x-5)

$\frac{7 x + 3}{x - 5}$

Step 1

Interchange the variables.

$x = \frac{7 y + 3}{y - 5}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 5$ .

$(y - 5) (x) = (\frac{7 y + 3}{y - 5}) (y - 5)$

Simplify the left side.

Apply the distributive property.

$y x - 5 x = (\frac{7 y + 3}{y - 5}) (y - 5)$

Simplify the right side.

Cancel the common factor of $y - 5$ .

Cancel the common factor.

$y x - 5 x = \frac{7 y + 3}{y - 5} (y - 5)$

Rewrite the expression.

$y x - 5 x = 7 y + 3$

Solve for $y$ .

Subtract $7 y$ from both sides of the equation.

$y x - 5 x - 7 y = 3$

Add $5 x$ to both sides of the equation.

$y x - 7 y = 3 + 5 x$

Factor $y$ out of $y x - 7 y$ .

Factor $y$ out of $y x$ .

$y (x) - 7 y = 3 + 5 x$

Factor $y$ out of $- 7 y$ .

$y (x) + y \cdot - 7 = 3 + 5 x$

Factor $y$ out of $y (x) + y \cdot - 7$ .

$y (x - 7) = 3 + 5 x$

Divide each term in $y (x - 7) = 3 + 5 x$ by $x - 7$ and simplify.

Divide each term in $y (x - 7) = 3 + 5 x$ by $x - 7$ .

$\frac{y (x - 7)}{x - 7} = \frac{3}{x - 7} + \frac{5 x}{x - 7}$

Simplify the left side.

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{y (x - 7)}{x - 7} = \frac{3}{x - 7} + \frac{5 x}{x - 7}$

Divide $y$ by $1$ .

$y = \frac{3}{x - 7} + \frac{5 x}{x - 7}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{3 + 5 x}{x - 7}$

Step 3

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

$f^{- 1} (x) = \frac{3 + 5 x}{x - 7}$

Step 4

Verify if $f^{- 1} (x) = \frac{3 + 5 x}{x - 7}$ is the inverse of $f (x) = \frac{7 x + 3}{x - 5}$ .

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 $f^{- 1} (\frac{7 x + 3}{x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{3 + 5 (\frac{7 x + 3}{x - 5})}{(\frac{7 x + 3}{x - 5}) - 7}$

Multiply the numerator and denominator of the complex fraction by $x - 5$ .

Multiply $\frac{3 + 5 \frac{7 x + 3}{x - 5}}{\frac{7 x + 3}{x - 5} - 7}$ by $\frac{x - 5}{x - 5}$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{x - 5}{x - 5} \cdot \frac{3 + 5 (\frac{7 x + 3}{x - 5})}{\frac{7 x + 3}{x - 5} - 7}$

Combine.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (3 + 5 (\frac{7 x + 3}{x - 5}))}{(x - 5) (\frac{7 x + 3}{x - 5} - 7)}$

Apply the distributive property.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) \cdot 3 + (x - 5) (5 (\frac{7 x + 3}{x - 5}))}{(x - 5) (\frac{7 x + 3}{x - 5}) + (x - 5) \cdot - 7}$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) \cdot 3 + (x - 5) (5 (\frac{7 x + 3}{x - 5}))}{(x - 5) (\frac{7 x + 3}{x - 5}) + (x - 5) \cdot - 7}$

Rewrite the expression.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) \cdot 3 + (x - 5) (5 (\frac{7 x + 3}{x - 5}))}{7 x + 3 + (x - 5) \cdot - 7}$

Simplify the numerator.

Factor $x - 5$ out of $(x - 5) \cdot 3 + (x - 5) \cdot 5 \frac{7 x + 3}{x - 5}$ .

Factor $x - 5$ out of $(x - 5) \cdot 3$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (3) + (x - 5) (5 (\frac{7 x + 3}{x - 5}))}{7 x + 3 + (x - 5) \cdot - 7}$

Factor $x - 5$ out of $(x - 5) (3) + (x - 5) (5 \frac{7 x + 3}{x - 5})$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (3 + 5 (\frac{7 x + 3}{x - 5}))}{7 x + 3 + (x - 5) \cdot - 7}$

Combine $5$ and $\frac{7 x + 3}{x - 5}$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (3 + \frac{5 (7 x + 3)}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{3 (x - 5)}{x - 5} + \frac{5 (7 x + 3)}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{3 (x - 5) + 5 (7 x + 3)}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Reorder terms.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{5 (7 x + 3) + 3 (x - 5)}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Rewrite $\frac{5 (7 x + 3) + 3 (x - 5)}{x - 5}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{5 (7 x) + 5 \cdot 3 + 3 (x - 5)}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Multiply $7$ by $5$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{35 x + 5 \cdot 3 + 3 (x - 5)}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Multiply $5$ by $3$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{35 x + 15 + 3 (x - 5)}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Apply the distributive property.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{35 x + 15 + 3 x + 3 \cdot - 5}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Multiply $3$ by $- 5$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{35 x + 15 + 3 x - 15}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Add $35 x$ and $3 x$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x + 15 - 15}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Subtract $15$ from $15$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x + 0}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Add $38 x$ and $0$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x}{x - 5})}{7 x + 3 + (x - 5) \cdot - 7}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x}{x - 5})}{7 x + 3 + x \cdot - 7 - 5 \cdot - 7}$

Move $- 7$ to the left of $x$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x}{x - 5})}{7 x + 3 - 7 \cdot x - 5 \cdot - 7}$

Multiply $- 5$ by $- 7$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x}{x - 5})}{7 x + 3 - 7 \cdot x + 35}$

Subtract $7 x$ from $7 x$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x}{x - 5})}{0 + 3 + 35}$

Add $0$ and $3$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x}{x - 5})}{3 + 35}$

Add $3$ and $35$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{(x - 5) (\frac{38 x}{x - 5})}{38}$

Reduce the expression by cancelling the common factors.

Factor $\frac{38 x}{x - 5}$ out of $\frac{(x - 5) \frac{38 x}{x - 5}}{38}$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{38 x}{x - 5} \cdot \frac{x - 5}{38}$

Cancel the common factor of $38$ .

Factor $38$ out of $38 x$ .

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{38 (x)}{x - 5} \cdot \frac{x - 5}{38}$

Cancel the common factor.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{38 x}{x - 5} \cdot \frac{x - 5}{38}$

Rewrite the expression.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Rewrite the expression.

$f^{- 1} (\frac{7 x + 3}{x - 5}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{3 + 5 x}{x - 7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{7 (\frac{3 + 5 x}{x - 7}) + 3}{(\frac{3 + 5 x}{x - 7}) - 5}$

Multiply the numerator and denominator of the complex fraction by $x - 7$ .

Multiply $\frac{7 \frac{3 + 5 x}{x - 7} + 3}{\frac{3 + 5 x}{x - 7} - 5}$ by $\frac{x - 7}{x - 7}$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{x - 7}{x - 7} \cdot \frac{7 (\frac{3 + 5 x}{x - 7}) + 3}{\frac{3 + 5 x}{x - 7} - 5}$

Combine.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{3 + 5 x}{x - 7}) + 3)}{(x - 7) (\frac{3 + 5 x}{x - 7} - 5)}$

Apply the distributive property.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{3 + 5 x}{x - 7})) + (x - 7) \cdot 3}{(x - 7) (\frac{3 + 5 x}{x - 7}) + (x - 7) \cdot - 5}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{3 + 5 x}{x - 7})) + (x - 7) \cdot 3}{(x - 7) (\frac{3 + 5 x}{x - 7}) + (x - 7) \cdot - 5}$

Rewrite the expression.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{3 + 5 x}{x - 7})) + (x - 7) \cdot 3}{3 + 5 x + (x - 7) \cdot - 5}$

Simplify the numerator.

Factor $x - 7$ out of $(x - 7) \cdot 7 \frac{3 + 5 x}{x - 7} + (x - 7) \cdot 3$ .

Factor $x - 7$ out of $(x - 7) \cdot 7 \frac{3 + 5 x}{x - 7}$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{3 + 5 x}{x - 7})) + (x - 7) \cdot 3}{3 + 5 x + (x - 7) \cdot - 5}$

Factor $x - 7$ out of $(x - 7) \cdot 3$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{3 + 5 x}{x - 7})) + (x - 7) (3)}{3 + 5 x + (x - 7) \cdot - 5}$

Factor $x - 7$ out of $(x - 7) (7 \frac{3 + 5 x}{x - 7}) + (x - 7) (3)$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (7 (\frac{3 + 5 x}{x - 7}) + 3)}{3 + 5 x + (x - 7) \cdot - 5}$

Combine $7$ and $\frac{3 + 5 x}{x - 7}$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 (3 + 5 x)}{x - 7} + 3)}{3 + 5 x + (x - 7) \cdot - 5}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 (3 + 5 x)}{x - 7} + \frac{3 (x - 7)}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Combine the numerators over the common denominator.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 (3 + 5 x) + 3 (x - 7)}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Rewrite $\frac{7 (3 + 5 x) + 3 (x - 7)}{x - 7}$ in a factored form.

Apply the distributive property.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{7 \cdot 3 + 7 (5 x) + 3 (x - 7)}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Multiply $7$ by $3$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{21 + 7 (5 x) + 3 (x - 7)}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Multiply $5$ by $7$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{21 + 35 x + 3 (x - 7)}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Apply the distributive property.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{21 + 35 x + 3 x + 3 \cdot - 7}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Multiply $3$ by $- 7$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{21 + 35 x + 3 x - 21}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Subtract $21$ from $21$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{35 x + 3 x + 0}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Add $35 x + 3 x$ and $0$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{35 x + 3 x}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Add $35 x$ and $3 x$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{38 x}{x - 7})}{3 + 5 x + (x - 7) \cdot - 5}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{38 x}{x - 7})}{3 + 5 x + x \cdot - 5 - 7 \cdot - 5}$

Move $- 5$ to the left of $x$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{38 x}{x - 7})}{3 + 5 x - 5 \cdot x - 7 \cdot - 5}$

Multiply $- 7$ by $- 5$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{38 x}{x - 7})}{3 + 5 x - 5 \cdot x + 35}$

Add $3$ and $35$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{38 x}{x - 7})}{5 x - 5 x + 38}$

Subtract $5 x$ from $5 x$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{38 x}{x - 7})}{0 + 38}$

Add $0$ and $38$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{(x - 7) (\frac{38 x}{x - 7})}{38}$

Reduce the expression by cancelling the common factors.

Factor $\frac{38 x}{x - 7}$ out of $\frac{(x - 7) \frac{38 x}{x - 7}}{38}$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{38 x}{x - 7} \cdot \frac{x - 7}{38}$

Cancel the common factor of $38$ .

Factor $38$ out of $38 x$ .

$f (\frac{3 + 5 x}{x - 7}) = \frac{38 (x)}{x - 7} \cdot \frac{x - 7}{38}$

Cancel the common factor.

$f (\frac{3 + 5 x}{x - 7}) = \frac{38 x}{x - 7} \cdot \frac{x - 7}{38}$

Rewrite the expression.

$f (\frac{3 + 5 x}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$f (\frac{3 + 5 x}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Rewrite the expression.

$f (\frac{3 + 5 x}{x - 7}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 + 5 x}{x - 7}$ is the inverse of $f (x) = \frac{7 x + 3}{x - 5}$ .

$f^{- 1} (x) = \frac{3 + 5 x}{x - 7}$

Find the Inverse (7x+3)/(x-5)

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