Calculus Examples

$f (x) = x + 5$

Step 1

Write $f (x) = x + 5$ as an equation.

$y = x + 5$

Step 2

Interchange the variables.

$x = y + 5$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y + 5 = x$ .

$y + 5 = x$

Step 3.2

Subtract $5$ from both sides of the equation.

$y = x - 5$

Step 4

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

$f^{- 1} (x) = x - 5$

Step 5

Verify if $f^{- 1} (x) = x - 5$ is the inverse of $f (x) = x + 5$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (x + 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x + 5) = (x + 5) - 5$

Step 5.2.3

Combine the opposite terms in $(x + 5) - 5$ .

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Step 5.2.3.1

Subtract $5$ from $5$ .

$f^{- 1} (x + 5) = x + 0$

Step 5.2.3.2

Add $x$ and $0$ .

$f^{- 1} (x + 5) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (x - 5)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x - 5) = (x - 5) + 5$

Step 5.3.3

Combine the opposite terms in $(x - 5) + 5$ .

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Step 5.3.3.1

Add $- 5$ and $5$ .

$f (x - 5) = x + 0$

Step 5.3.3.2

Add $x$ and $0$ .

$f (x - 5) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x - 5$ is the inverse of $f (x) = x + 5$ .

$f^{- 1} (x) = x - 5$

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