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

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

Move the negative in front of the fraction.

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

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.

$y + 4, 1$

Remove parentheses.

$y + 4, 1$

The LCM of one and any expression is the expression.

$y + 4$

Multiply each term in $- \frac{3}{y + 4} = x$ by $y + 4$ to eliminate the fractions.

Multiply each term in $- \frac{3}{y + 4} = x$ by $y + 4$ .

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

Simplify the left side.

Cancel the common factor of $y + 4$ .

Move the leading negative in $- \frac{3}{y + 4}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

$- 3 = x (y + 4)$

Simplify the right side.

Apply the distributive property.

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

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

$- 3 = x y + 4 x$

Solve the equation.

Rewrite the equation as $x y + 4 x = - 3$ .

$x y + 4 x = - 3$

Subtract $4 x$ from both sides of the equation.

$x y = - 3 - 4 x$

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

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

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $- 4$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative one from the denominator of $\frac{x + 4}{- 1}$ .

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

Rewrite $- 1 \cdot (x + 4)$ as $- (x + 4)$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $4$ .

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

Apply the distributive property.

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $- 4$ .

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

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

Subtract $4$ from $4$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify the denominator.

Add $- 4$ and $4$ .

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{x}{3}$ into the numerator.

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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