Find the Inverse f(x)=(8x+5)/3

$f (x) = \frac{8 x + 5}{3}$

Step 1

Write $f (x) = \frac{8 x + 5}{3}$ as an equation.

$y = \frac{8 x + 5}{3}$

Step 2

Interchange the variables.

$x = \frac{8 y + 5}{3}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{8 y + 5}{3} = x$ .

$\frac{8 y + 5}{3} = x$

Multiply both sides by $3$ .

$\frac{8 y + 5}{3} \cdot 3 = x \cdot 3$

Simplify.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{8 y + 5}{3} \cdot 3 = x \cdot 3$

Rewrite the expression.

$8 y + 5 = x \cdot 3$

Simplify the right side.

Move $3$ to the left of $x$ .

$8 y + 5 = 3 x$

Solve for $y$ .

Subtract $5$ from both sides of the equation.

$8 y = 3 x - 5$

Divide each term in $8 y = 3 x - 5$ by $8$ and simplify.

Divide each term in $8 y = 3 x - 5$ by $8$ .

$\frac{8 y}{8} = \frac{3 x}{8} + \frac{- 5}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{3 x}{8} + \frac{- 5}{8}$

Divide $y$ by $1$ .

$y = \frac{3 x}{8} + \frac{- 5}{8}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{3 x}{8} - \frac{5}{8}$

Step 4

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

$f^{- 1} (x) = \frac{3 x}{8} - \frac{5}{8}$

Step 5

Verify if $f^{- 1} (x) = \frac{3 x}{8} - \frac{5}{8}$ is the inverse of $f (x) = \frac{8 x + 5}{3}$ .

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 $f^{- 1} (\frac{8 x + 5}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{8 x + 5}{3}) = \frac{3 (\frac{8 x + 5}{3})}{8} - \frac{5}{8}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{8 x + 5}{3}) = \frac{3 (\frac{8 x + 5}{3}) - 5}{8}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{8 x + 5}{3}) = \frac{3 (\frac{8 x + 5}{3}) - 5}{8}$

Rewrite the expression.

$f^{- 1} (\frac{8 x + 5}{3}) = \frac{8 x + 5 - 5}{8}$

Combine the opposite terms in $8 x + 5 - 5$ .

Subtract $5$ from $5$ .

$f^{- 1} (\frac{8 x + 5}{3}) = \frac{8 x + 0}{8}$

Add $8 x$ and $0$ .

$f^{- 1} (\frac{8 x + 5}{3}) = \frac{8 x}{8}$

Cancel the common factor of $8$ .

Cancel the common factor.

$f^{- 1} (\frac{8 x + 5}{3}) = \frac{8 x}{8}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{8 x + 5}{3}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{3 x}{8} - \frac{5}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{8 (\frac{3 x}{8} - \frac{5}{8}) + 5}{3}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{8 (\frac{3 x}{8}) + 8 (- \frac{5}{8}) + 5}{3}$

Cancel the common factor of $8$ .

Cancel the common factor.

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{8 (\frac{3 x}{8}) + 8 (- \frac{5}{8}) + 5}{3}$

Rewrite the expression.

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{3 x + 8 (- \frac{5}{8}) + 5}{3}$

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{3 x + 8 (\frac{- 5}{8}) + 5}{3}$

Cancel the common factor.

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{3 x + 8 (\frac{- 5}{8}) + 5}{3}$

Rewrite the expression.

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{3 x - 5 + 5}{3}$

Add $- 5$ and $5$ .

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{3 x}{8} - \frac{5}{8}) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$f (\frac{3 x}{8} - \frac{5}{8}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{8} - \frac{5}{8}$ is the inverse of $f (x) = \frac{8 x + 5}{3}$ .

$f^{- 1} (x) = \frac{3 x}{8} - \frac{5}{8}$

Find the Inverse f(x)=(8x+5)/3

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