Find the Inverse 2((x-7)/2)-7

$2 (\frac{x - 7}{2}) - 7$

Step 1

Interchange the variables.

$x = 2 (\frac{y - 7}{2}) - 7$

Step 2

Solve for $y$ .

Rewrite the equation as $2 (\frac{y - 7}{2}) - 7 = x$ .

$2 (\frac{y - 7}{2}) - 7 = x$

Simplify $2 (\frac{y - 7}{2}) - 7$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{y - 7}{2} - 7 = x$

Rewrite the expression.

$y - 7 - 7 = x$

Subtract $7$ from $- 7$ .

$y - 14 = x$

Add $14$ to both sides of the equation.

$y = x + 14$

Step 3

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

$f^{- 1} (x) = x + 14$

Step 4

Verify if $f^{- 1} (x) = x + 14$ is the inverse of $f (x) = 2 (\frac{x - 7}{2}) - 7$ .

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 $f^{- 1} (2 (\frac{x - 7}{2}) - 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 (\frac{x - 7}{2}) - 7) = (2 (\frac{x - 7}{2}) - 7) + 14$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (2 (\frac{x - 7}{2}) - 7) = 2 (\frac{x - 7}{2}) - 7 + 14$

Rewrite the expression.

$f^{- 1} (2 (\frac{x - 7}{2}) - 7) = x - 7 - 7 + 14$

Subtract $7$ from $- 7$ .

$f^{- 1} (2 (\frac{x - 7}{2}) - 7) = x - 14 + 14$

Combine the opposite terms in $x - 14 + 14$ .

Add $- 14$ and $14$ .

$f^{- 1} (2 (\frac{x - 7}{2}) - 7) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (2 (\frac{x - 7}{2}) - 7) = x$

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

Set up the composite result function.

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

Evaluate $f (x + 14)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x + 14) = 2 (\frac{(x + 14) - 7}{2}) - 7$

Simplify each term.

Subtract $7$ from $14$ .

$f (x + 14) = 2 (\frac{x + 7}{2}) - 7$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (x + 14) = 2 (\frac{x + 7}{2}) - 7$

Rewrite the expression.

$f (x + 14) = x + 7 - 7$

Combine the opposite terms in $x + 7 - 7$ .

Subtract $7$ from $7$ .

$f (x + 14) = x + 0$

Add $x$ and $0$ .

$f (x + 14) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x + 14$ is the inverse of $f (x) = 2 (\frac{x - 7}{2}) - 7$ .

$f^{- 1} (x) = x + 14$

Find the Inverse 2((x-7)/2)-7

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