Find the Inverse f^-1(10)

Divide each term in ${\displaystyle (y^{-1}\cdot 10)=f}$ by ${\displaystyle 10}$.

Simplify the left side.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle y,10}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,10}$ then find LCM for the variable part ${\displaystyle y^1}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

${\displaystyle 10}$ has factors of ${\displaystyle 2}$ and ${\displaystyle 5}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle 5}$.

The factor for ${\displaystyle y^1}$ is ${\displaystyle y}$ itself.

The LCM of ${\displaystyle y^1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

The LCM for ${\displaystyle y,10}$ is the numeric part ${\displaystyle 10}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{1}{y}=\frac{f}{10}}$ by ${\displaystyle 10y}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle fy=10}$.

Divide each term in ${\displaystyle fy=10}$ by ${\displaystyle f}$ and simplify.

Set up the composite result function.

Evaluate ${\displaystyle f^{-1}\left(f^{-1}\left(10\right)\right)}$ by substituting in the value of ${\displaystyle f}$ into ${\displaystyle f^{-1}}$.

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

Cancel the common factor of ${\displaystyle 10}$.

Set up the composite result function.

Evaluate ${\displaystyle f\left(\frac{10}{f}\right)}$ by substituting in the value of ${\displaystyle f^{-1}}$ into ${\displaystyle f}$.

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

Cancel the common factor of ${\displaystyle 10}$.