Find the Inverse f(x)=2/x

$f (x) = \frac{2}{x}$

Step 1

Write $f (x) = \frac{2}{x}$ as an equation.

$y = \frac{2}{x}$

Step 2

Interchange the variables.

$x = \frac{2}{y}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{2}{y} = x$ .

$\frac{2}{y} = x$

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 $\frac{2}{y} = x$ by $y$ to eliminate the fractions.

Multiply each term in $\frac{2}{y} = x$ by $y$ .

$\frac{2}{y} y = x y$

Simplify the left side.

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{2}{y} y = x y$

Rewrite the expression.

$2 = x y$

Solve the equation.

Rewrite the equation as $x y = 2$ .

$x y = 2$

Divide each term in $x y = 2$ by $x$ and simplify.

Divide each term in $x y = 2$ by $x$ .

$\frac{x y}{x} = \frac{2}{x}$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y}{x} = \frac{2}{x}$

Divide $y$ by $1$ .

$y = \frac{2}{x}$

Step 4

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

$f^{- 1} (x) = \frac{2}{x}$

Step 5

Verify if $f^{- 1} (x) = \frac{2}{x}$ is the inverse of $f (x) = \frac{2}{x}$ .

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 $f^{- 1} (\frac{2}{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2}{x}) = \frac{2}{\frac{2}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{2}{x}) = 2 (\frac{x}{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{2}{x}) = 2 (\frac{x}{2})$

Rewrite the expression.

$f^{- 1} (\frac{2}{x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{2}{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2}{x}) = \frac{2}{\frac{2}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2}{x}) = 2 (\frac{x}{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{2}{x}) = 2 (\frac{x}{2})$

Rewrite the expression.

$f (\frac{2}{x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2}{x}$ is the inverse of $f (x) = \frac{2}{x}$ .

$f^{- 1} (x) = \frac{2}{x}$

Find the Inverse f(x)=2/x

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