Find the Inverse f(x)=-3/x

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

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.

$y, 1$

The LCM of one and any expression is the expression.

$y$

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

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

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

Simplify the left side.

Cancel the common factor of $y$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$- 3 = x y$

Solve the equation.

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

$x y = - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Set up the composite result function.

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Find the Inverse f(x)=-3/x

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