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

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

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

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

Add $4$ to both sides of the equation.

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

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

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

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $7$ .

Factor $7$ out of $7 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

Apply the distributive property.

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

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

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

Multiply $\frac{2}{7} \cdot 4$ .

Combine $\frac{2}{7}$ and $4$ .

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

Multiply $2$ by $4$ .

$y = \frac{2 x}{7} + \frac{8}{7}$

Step 4

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

$f^{- 1} (x) = \frac{2 x}{7} + \frac{8}{7}$

Step 5

Verify if $f^{- 1} (x) = \frac{2 x}{7} + \frac{8}{7}$ is the inverse of $f (x) = - 4 + \frac{7}{2} x$ .

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

$f^{- 1} (- 4 + \frac{7}{2} x) = \frac{2 (- 4 + \frac{7}{2} x)}{7} + \frac{8}{7}$

Combine the numerators over the common denominator.

$f^{- 1} (- 4 + \frac{7}{2} x) = \frac{2 (- 4 + \frac{7}{2} x) + 8}{7}$

Simplify each term.

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

$f^{- 1} (- 4 + \frac{7}{2} x) = \frac{2 (- 4 + \frac{7 x}{2}) + 8}{7}$

Apply the distributive property.

$f^{- 1} (- 4 + \frac{7}{2} x) = \frac{2 \cdot - 4 + 2 (\frac{7 x}{2}) + 8}{7}$

Multiply $2$ by $- 4$ .

$f^{- 1} (- 4 + \frac{7}{2} x) = \frac{- 8 + 2 (\frac{7 x}{2}) + 8}{7}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (- 4 + \frac{7}{2} x) = \frac{- 8 + 2 (\frac{7 x}{2}) + 8}{7}$

Rewrite the expression.

$f^{- 1} (- 4 + \frac{7}{2} x) = \frac{- 8 + 7 x + 8}{7}$

Simplify terms.

Combine the opposite terms in $- 8 + 7 x + 8$ .

Add $- 8$ and $8$ .

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

Add $7 x$ and $0$ .

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

Cancel the common factor of $7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{2 x}{7} + \frac{8}{7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + \frac{7}{2} \cdot (\frac{2 x}{7} + \frac{8}{7})$

Simplify each term.

Apply the distributive property.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + \frac{7}{2} \cdot \frac{2 x}{7} + \frac{7}{2} \cdot \frac{8}{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + \frac{7}{2} \cdot \frac{2 x}{7} + \frac{7}{2} \cdot \frac{8}{7}$

Rewrite the expression.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + \frac{1}{2} \cdot (2 x) + \frac{7}{2} \cdot \frac{8}{7}$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x$ .

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + \frac{1}{2} \cdot (2 (x)) + \frac{7}{2} \cdot \frac{8}{7}$

Cancel the common factor.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + \frac{1}{2} \cdot (2 x) + \frac{7}{2} \cdot \frac{8}{7}$

Rewrite the expression.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + x + \frac{7}{2} \cdot \frac{8}{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + x + \frac{7}{2} \cdot \frac{8}{7}$

Rewrite the expression.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + x + \frac{1}{2} \cdot 8$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + x + \frac{1}{2} \cdot (2 (4))$

Cancel the common factor.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + x + \frac{1}{2} \cdot (2 \cdot 4)$

Rewrite the expression.

$f (\frac{2 x}{7} + \frac{8}{7}) = - 4 + x + 4$

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

Add $- 4$ and $4$ .

$f (\frac{2 x}{7} + \frac{8}{7}) = x + 0$

Add $x$ and $0$ .

$f (\frac{2 x}{7} + \frac{8}{7}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 x}{7} + \frac{8}{7}$ is the inverse of $f (x) = - 4 + \frac{7}{2} x$ .

$f^{- 1} (x) = \frac{2 x}{7} + \frac{8}{7}$

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

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