Find the Inverse f(x)=7-2x

$f (x) = 7 - 2 x$

Step 1

Write $f (x) = 7 - 2 x$ as an equation.

$y = 7 - 2 x$

Step 2

Interchange the variables.

$x = 7 - 2 y$

Step 3

Solve for $y$ .

Rewrite the equation as $7 - 2 y = x$ .

$7 - 2 y = x$

Subtract $7$ from both sides of the equation.

$- 2 y = x - 7$

Divide each term in $- 2 y = x - 7$ by $- 2$ and simplify.

Divide each term in $- 2 y = x - 7$ by $- 2$ .

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Dividing two negative values results in a positive value.

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

Step 4

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

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

Step 5

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $- 1$ by $7$ .

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

Multiply $- 2$ by $- 1$ .

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

Simplify terms.

Combine the opposite terms in $- 7 + 2 x + 7$ .

Add $- 7$ and $7$ .

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

Add $2 x$ and $0$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2}$ into the numerator.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $7$ .

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

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

Subtract $7$ from $7$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=7-2x

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