Find the Inverse f(x)=(5x+1)/7

$f (x) = \frac{5 x + 1}{7}$

Step 1

Write $f (x) = \frac{5 x + 1}{7}$ as an equation.

$y = \frac{5 x + 1}{7}$

Step 2

Interchange the variables.

$x = \frac{5 y + 1}{7}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{5 y + 1}{7} = x$ .

$\frac{5 y + 1}{7} = x$

Multiply both sides by $7$ .

$\frac{5 y + 1}{7} \cdot 7 = x \cdot 7$

Simplify.

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{5 y + 1}{7} \cdot 7 = x \cdot 7$

Rewrite the expression.

$5 y + 1 = x \cdot 7$

Simplify the right side.

Move $7$ to the left of $x$ .

$5 y + 1 = 7 x$

Solve for $y$ .

Subtract $1$ from both sides of the equation.

$5 y = 7 x - 1$

Divide each term in $5 y = 7 x - 1$ by $5$ and simplify.

Divide each term in $5 y = 7 x - 1$ by $5$ .

$\frac{5 y}{5} = \frac{7 x}{5} + \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{7 x}{5} + \frac{- 1}{5}$

Divide $y$ by $1$ .

$y = \frac{7 x}{5} + \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{7 x}{5} - \frac{1}{5}$

Step 4

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

$f^{- 1} (x) = \frac{7 x}{5} - \frac{1}{5}$

Step 5

Verify if $f^{- 1} (x) = \frac{7 x}{5} - \frac{1}{5}$ is the inverse of $f (x) = \frac{5 x + 1}{7}$ .

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 $f^{- 1} (\frac{5 x + 1}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x + 1}{7}) = \frac{7 (\frac{5 x + 1}{7})}{5} - \frac{1}{5}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x + 1}{7}) = \frac{7 (\frac{5 x + 1}{7}) - 1}{5}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (\frac{5 x + 1}{7}) = \frac{7 (\frac{5 x + 1}{7}) - 1}{5}$

Rewrite the expression.

$f^{- 1} (\frac{5 x + 1}{7}) = \frac{5 x + 1 - 1}{5}$

Combine the opposite terms in $5 x + 1 - 1$ .

Subtract $1$ from $1$ .

$f^{- 1} (\frac{5 x + 1}{7}) = \frac{5 x + 0}{5}$

Add $5 x$ and $0$ .

$f^{- 1} (\frac{5 x + 1}{7}) = \frac{5 x}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (\frac{5 x + 1}{7}) = \frac{5 x}{5}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{5 x + 1}{7}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{7 x}{5} - \frac{1}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{5 (\frac{7 x}{5} - \frac{1}{5}) + 1}{7}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{5 (\frac{7 x}{5}) + 5 (- \frac{1}{5}) + 1}{7}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{5 (\frac{7 x}{5}) + 5 (- \frac{1}{5}) + 1}{7}$

Rewrite the expression.

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{7 x + 5 (- \frac{1}{5}) + 1}{7}$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{7 x + 5 (\frac{- 1}{5}) + 1}{7}$

Cancel the common factor.

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{7 x + 5 (\frac{- 1}{5}) + 1}{7}$

Rewrite the expression.

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{7 x - 1 + 1}{7}$

Add $- 1$ and $1$ .

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{7 x + 0}{7}$

Add $7 x$ and $0$ .

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{7 x}{7}$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{7 x}{5} - \frac{1}{5}) = \frac{7 x}{7}$

Divide $x$ by $1$ .

$f (\frac{7 x}{5} - \frac{1}{5}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{7 x}{5} - \frac{1}{5}$ is the inverse of $f (x) = \frac{5 x + 1}{7}$ .

$f^{- 1} (x) = \frac{7 x}{5} - \frac{1}{5}$

Find the Inverse f(x)=(5x+1)/7

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