Find the Inverse f^-1

$f^{- 1}$

Step 1

Interchange the variables.

$f = y^{- 1}$

Step 2

Solve for $y$ .

Rewrite the equation as $y^{- 1} = f$ .

$y^{- 1} = f$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{y} = f$

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 $\frac{1}{y} = f$ by $y$ to eliminate the fractions.

Multiply each term in $\frac{1}{y} = f$ by $y$ .

$\frac{1}{y} y = f y$

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{1}{y} y = f y$

Rewrite the expression.

$1 = f y$

Solve the equation.

Rewrite the equation as $f y = 1$ .

$f y = 1$

Divide each term in $f y = 1$ by $f$ and simplify.

Divide each term in $f y = 1$ by $f$ .

$\frac{f y}{f} = \frac{1}{f}$

Simplify the left side.

Cancel the common factor of $f$ .

Cancel the common factor.

$\frac{f y}{f} = \frac{1}{f}$

Divide $y$ by $1$ .

$y = \frac{1}{f}$

Step 3

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

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

Step 4

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

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

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

Set up the composite result function.

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

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

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

Move $f^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

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

Set up the composite result function.

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

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

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

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (\frac{1}{f}) = f$

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

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

Find the Inverse f^-1

Related Questions

Support LCMGCF.com by disabling your adblocker.