Find the Inverse f(x)=1/3-1/21x

$f (x) = \frac{1}{3} - \frac{1}{21} x$

Step 1

Write $f (x) = \frac{1}{3} - \frac{1}{21} x$ as an equation.

$y = \frac{1}{3} - \frac{1}{21} x$

Step 2

Interchange the variables.

$x = \frac{1}{3} - \frac{1}{21} y$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{1}{3} - \frac{1}{21} y = x$ .

$\frac{1}{3} - \frac{1}{21} y = x$

Combine $y$ and $\frac{1}{21}$ .

$\frac{1}{3} - \frac{y}{21} = x$

Subtract $\frac{1}{3}$ from both sides of the equation.

$- \frac{y}{21} = x - \frac{1}{3}$

Multiply both sides of the equation by $- 21$ .

$- 21 (- \frac{y}{21}) = - 21 (x - \frac{1}{3})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 21 (- \frac{y}{21})$ .

Cancel the common factor of $21$ .

Move the leading negative in $- \frac{y}{21}$ into the numerator.

$- 21 \frac{- y}{21} = - 21 (x - \frac{1}{3})$

Factor $21$ out of $- 21$ .

$21 (- 1) \frac{- y}{21} = - 21 (x - \frac{1}{3})$

Cancel the common factor.

$21 \cdot - 1 \frac{- y}{21} = - 21 (x - \frac{1}{3})$

Rewrite the expression.

$- - y = - 21 (x - \frac{1}{3})$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - 21 (x - \frac{1}{3})$

Multiply $y$ by $1$ .

$y = - 21 (x - \frac{1}{3})$

Simplify the right side.

Simplify $- 21 (x - \frac{1}{3})$ .

Apply the distributive property.

$y = - 21 x - 21 (- \frac{1}{3})$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y = - 21 x - 21 (\frac{- 1}{3})$

Factor $3$ out of $- 21$ .

$y = - 21 x + 3 (- 7) \frac{- 1}{3}$

Cancel the common factor.

$y = - 21 x + 3 \cdot - 7 \frac{- 1}{3}$

Rewrite the expression.

$y = - 21 x - 7 \cdot - 1$

Multiply $- 7$ by $- 1$ .

$y = - 21 x + 7$

Step 4

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

$f^{- 1} (x) = - 21 x + 7$

Step 5

Verify if $f^{- 1} (x) = - 21 x + 7$ is the inverse of $f (x) = \frac{1}{3} - \frac{1}{21} x$ .

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 $f^{- 1} (\frac{1}{3} - \frac{1}{21} x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 21 (\frac{1}{3} - \frac{1}{21} x) + 7$

Simplify each term.

Combine $x$ and $\frac{1}{21}$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 21 (\frac{1}{3} - \frac{x}{21}) + 7$

Apply the distributive property.

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 21 (\frac{1}{3}) - 21 (- \frac{x}{21}) + 7$

Cancel the common factor of $3$ .

Factor $3$ out of $- 21$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = 3 (- 7) (\frac{1}{3}) - 21 (- \frac{x}{21}) + 7$

Cancel the common factor.

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = 3 \cdot (- 7 (\frac{1}{3})) - 21 (- \frac{x}{21}) + 7$

Rewrite the expression.

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 7 - 21 (- \frac{x}{21}) + 7$

Cancel the common factor of $21$ .

Move the leading negative in $- \frac{x}{21}$ into the numerator.

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 7 - 21 \frac{- x}{21} + 7$

Factor $21$ out of $- 21$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 7 + 21 (- 1) (\frac{- x}{21}) + 7$

Cancel the common factor.

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 7 + 21 \cdot (- 1 \frac{- x}{21}) + 7$

Rewrite the expression.

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 7 - 1 (- x) + 7$

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 7 + 1 x + 7$

Multiply $x$ by $1$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = - 7 + x + 7$

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

Add $- 7$ and $7$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{1}{3} - \frac{1}{21} x) = x$

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

Set up the composite result function.

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

Evaluate $f (- 21 x + 7)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- 21 x + 7) = \frac{1}{3} - \frac{1}{21} \cdot (- 21 x + 7)$

Simplify each term.

Apply the distributive property.

$f (- 21 x + 7) = \frac{1}{3} - \frac{1}{21} \cdot (- 21 x) - \frac{1}{21} \cdot 7$

Cancel the common factor of $21$ .

Move the leading negative in $- \frac{1}{21}$ into the numerator.

$f (- 21 x + 7) = \frac{1}{3} + \frac{- 1}{21} \cdot (- 21 x) - \frac{1}{21} \cdot 7$

Factor $21$ out of $- 21 x$ .

$f (- 21 x + 7) = \frac{1}{3} + \frac{- 1}{21} \cdot (21 (- x)) - \frac{1}{21} \cdot 7$

Cancel the common factor.

$f (- 21 x + 7) = \frac{1}{3} + \frac{- 1}{21} \cdot (21 (- x)) - \frac{1}{21} \cdot 7$

Rewrite the expression.

$f (- 21 x + 7) = \frac{1}{3} - 1 (- x) - \frac{1}{21} \cdot 7$

Multiply $- 1$ by $- 1$ .

$f (- 21 x + 7) = \frac{1}{3} + 1 x - \frac{1}{21} \cdot 7$

Multiply $x$ by $1$ .

$f (- 21 x + 7) = \frac{1}{3} + x - \frac{1}{21} \cdot 7$

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{1}{21}$ into the numerator.

$f (- 21 x + 7) = \frac{1}{3} + x + \frac{- 1}{21} \cdot 7$

Factor $7$ out of $21$ .

$f (- 21 x + 7) = \frac{1}{3} + x + \frac{- 1}{7 (3)} \cdot 7$

Cancel the common factor.

$f (- 21 x + 7) = \frac{1}{3} + x + \frac{- 1}{7 \cdot 3} \cdot 7$

Rewrite the expression.

$f (- 21 x + 7) = \frac{1}{3} + x + \frac{- 1}{3}$

Move the negative in front of the fraction.

$f (- 21 x + 7) = \frac{1}{3} + x - \frac{1}{3}$

Combine the opposite terms in $\frac{1}{3} + x - \frac{1}{3}$ .

Combine the numerators over the common denominator.

$f (- 21 x + 7) = x + \frac{1 - 1}{3}$

Subtract $1$ from $1$ .

$f (- 21 x + 7) = x + \frac{0}{3}$

Divide $0$ by $3$ .

$f (- 21 x + 7) = x + 0$

Add $x$ and $0$ .

$f (- 21 x + 7) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - 21 x + 7$ is the inverse of $f (x) = \frac{1}{3} - \frac{1}{21} x$ .

$f^{- 1} (x) = - 21 x + 7$

Find the Inverse f(x)=1/3-1/21x

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