Find the Inverse f(x)=-3x+21

$f (x) = - 3 x + 21$

Write $f (x) = - 3 x + 21$ as an equation.

$y = - 3 x + 21$

Interchange the variables.

$x = - 3 y + 21$

Solve for $y$ .

Rewrite the equation as $- 3 y + 21 = x$ .

$- 3 y + 21 = x$

Subtract $21$ from both sides of the equation.

$- 3 y = x - 21$

Divide each term in $- 3 y = x - 21$ by $- 3$ and simplify.

Divide each term in $- 3 y = x - 21$ by $- 3$ .

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Divide $- 21$ by $- 3$ .

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

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

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

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

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

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

Simplify each term.

Cancel the common factor of $- 3 x + 21$ and $3$ .

Factor $3$ out of $- 3 x$ .

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

Factor $3$ out of $21$ .

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

Factor $3$ out of $3 (- x) + 3 (7)$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x + 7$ by $1$ .

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

Apply the distributive property.

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $7$ .

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

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

Add $- 7$ and $7$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (- \frac{x}{3} + 7)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 3$ by $7$ .

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

Combine the opposite terms in $x - 21 + 21$ .

Add $- 21$ and $21$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=-3x+21

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