Calculus Examples

$2 x$

Step 1

Interchange the variables.

$x = 2 y$

Step 2

Solve for $y$ .

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Rewrite the equation as $2 y = x$ .

$2 y = x$

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

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

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

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Set up the composite result function.

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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Set up the composite result function.

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

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

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