Find the Inverse f(x)=(2x)/7+4

$f (x) = \frac{2 x}{7} + 4$

Step 1

Write $f (x) = \frac{2 x}{7} + 4$ as an equation.

$y = \frac{2 x}{7} + 4$

Step 2

Interchange the variables.

$x = \frac{2 y}{7} + 4$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{2 y}{7} + 4 = x$ .

$\frac{2 y}{7} + 4 = x$

Subtract $4$ from both sides of the equation.

$\frac{2 y}{7} = x - 4$

Multiply both sides of the equation by $\frac{7}{2}$ .

$\frac{7}{2} \cdot \frac{2 y}{7} = \frac{7}{2} (x - 4)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{7}{2} \cdot \frac{2 y}{7}$ .

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7}{2} \cdot \frac{2 y}{7} = \frac{7}{2} (x - 4)$

Rewrite the expression.

$\frac{1}{2} (2 y) = \frac{7}{2} (x - 4)$

Cancel the common factor of $2$ .

Factor $2$ out of $2 y$ .

$\frac{1}{2} (2 (y)) = \frac{7}{2} (x - 4)$

Cancel the common factor.

$\frac{1}{2} (2 y) = \frac{7}{2} (x - 4)$

Rewrite the expression.

$y = \frac{7}{2} (x - 4)$

Simplify the right side.

Simplify $\frac{7}{2} (x - 4)$ .

Apply the distributive property.

$y = \frac{7}{2} x + \frac{7}{2} \cdot - 4$

Combine $\frac{7}{2}$ and $x$ .

$y = \frac{7 x}{2} + \frac{7}{2} \cdot - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $7$ by $- 2$ .

$y = \frac{7 x}{2} - 14$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{7 x}{2} - 14$ is the inverse of $f (x) = \frac{2 x}{7} + 4$ .

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 $f^{- 1} (\frac{2 x}{7} + 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x}{7} + 4) = \frac{7 (\frac{2 x}{7} + 4)}{2} - 14$

Simplify each term.

Cancel the common factor of $\frac{2 x}{7} + 4$ and $2$ .

Factor $2$ out of $7 (\frac{2 x}{7} + 4)$ .

$f^{- 1} (\frac{2 x}{7} + 4) = \frac{2 (7 (\frac{x}{7} + 2))}{2} - 14$

Cancel the common factors.

Factor $2$ out of $2$ .

$f^{- 1} (\frac{2 x}{7} + 4) = \frac{2 (7 (\frac{x}{7} + 2))}{2 (1)} - 14$

Cancel the common factor.

$f^{- 1} (\frac{2 x}{7} + 4) = \frac{2 (7 (\frac{x}{7} + 2))}{2 \cdot 1} - 14$

Rewrite the expression.

$f^{- 1} (\frac{2 x}{7} + 4) = \frac{7 (\frac{x}{7} + 2)}{1} - 14$

Divide $7 (\frac{x}{7} + 2)$ by $1$ .

$f^{- 1} (\frac{2 x}{7} + 4) = 7 (\frac{x}{7} + 2) - 14$

Apply the distributive property.

$f^{- 1} (\frac{2 x}{7} + 4) = 7 (\frac{x}{7}) + 7 \cdot 2 - 14$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (\frac{2 x}{7} + 4) = 7 (\frac{x}{7}) + 7 \cdot 2 - 14$

Rewrite the expression.

$f^{- 1} (\frac{2 x}{7} + 4) = x + 7 \cdot 2 - 14$

Multiply $7$ by $2$ .

$f^{- 1} (\frac{2 x}{7} + 4) = x + 14 - 14$

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

Subtract $14$ from $14$ .

$f^{- 1} (\frac{2 x}{7} + 4) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{2 x}{7} + 4) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{7 x}{2} - 14)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x}{2} - 14) = \frac{2 (\frac{7 x}{2} - 14)}{7} + 4$

Simplify each term.

Cancel the common factor of $\frac{7 x}{2} - 14$ and $7$ .

Factor $7$ out of $2 (\frac{7 x}{2} - 14)$ .

$f (\frac{7 x}{2} - 14) = \frac{7 (2 (\frac{x}{2} - 2))}{7} + 4$

Cancel the common factors.

Factor $7$ out of $7$ .

$f (\frac{7 x}{2} - 14) = \frac{7 (2 (\frac{x}{2} - 2))}{7 (1)} + 4$

Cancel the common factor.

$f (\frac{7 x}{2} - 14) = \frac{7 (2 (\frac{x}{2} - 2))}{7 \cdot 1} + 4$

Rewrite the expression.

$f (\frac{7 x}{2} - 14) = \frac{2 (\frac{x}{2} - 2)}{1} + 4$

Divide $2 (\frac{x}{2} - 2)$ by $1$ .

$f (\frac{7 x}{2} - 14) = 2 (\frac{x}{2} - 2) + 4$

Apply the distributive property.

$f (\frac{7 x}{2} - 14) = 2 (\frac{x}{2}) + 2 \cdot - 2 + 4$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{7 x}{2} - 14) = 2 (\frac{x}{2}) + 2 \cdot - 2 + 4$

Rewrite the expression.

$f (\frac{7 x}{2} - 14) = x + 2 \cdot - 2 + 4$

Multiply $2$ by $- 2$ .

$f (\frac{7 x}{2} - 14) = x - 4 + 4$

Combine the opposite terms in $x - 4 + 4$ .

Add $- 4$ and $4$ .

$f (\frac{7 x}{2} - 14) = x + 0$

Add $x$ and $0$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{2} - 14$ is the inverse of $f (x) = \frac{2 x}{7} + 4$ .

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

Find the Inverse f(x)=(2x)/7+4

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