Find the Inverse f^-1(x)=1/3x+5/3

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

Write $f^{- 1} (x) = \frac{1}{3} x + \frac{5}{3}$ as an equation.

$y = \frac{1}{3} x + \frac{5}{3}$

Interchange the variables.

$x = \frac{1}{3} y + \frac{5}{3}$

Solve for $y$ .

Rewrite the equation as $\frac{1}{3} y + \frac{5}{3} = x$ .

$\frac{1}{3} y + \frac{5}{3} = x$

Combine $\frac{1}{3}$ and $y$ .

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

Subtract $\frac{5}{3}$ from both sides of the equation.

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

Multiply both sides of the equation by $3$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

Simplify $3 (x - \frac{5}{3})$ .

Apply the distributive property.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{5}{3}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

$y = 3 x - 5$

Replace $y$ with $f (x)$ to show the final answer.

$f (x) = 3 x - 5$

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

To verify the inverse, check if $f (f^{- 1} (x)) = x$ and $f^{- 1} (f (x)) = x$ .

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

Set up the composite result function.

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

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

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

Simplify each term.

Combine $\frac{1}{3}$ and $x$ .

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $5$ from $5$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $\frac{1}{3}$ and $- 5$ .

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

Move the negative in front of the fraction.

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

Combine the opposite terms in $x - \frac{5}{3} + \frac{5}{3}$ .

Add $- \frac{5}{3}$ and $\frac{5}{3}$ .

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

Add $x$ and $0$ .

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

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

$f (x) = 3 x - 5$

Find the Inverse f^-1(x)=1/3x+5/3

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