Precalculus Examples

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

Step 1

Write $f (x) = \frac{3 - x}{7}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3 - y}{7} = x$ .

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

Step 3.2

Multiply both sides by $7$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{3 - y}{7} \cdot 7$ .

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Step 3.3.1.1.1

Cancel the common factor of $7$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

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

Step 3.3.1.1.1.2

Rewrite the expression.

$3 - y = x \cdot 7$

Step 3.3.1.1.2

Reorder $3$ and $- y$ .

$- y + 3 = x \cdot 7$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

$- y + 3 = 7 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $3$ from both sides of the equation.

$- y = 7 x - 3$

Step 3.4.2

Divide each term in $- y = 7 x - 3$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- y = 7 x - 3$ by $- 1$ .

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.4.2.2.2

Divide $y$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Move the negative one from the denominator of $\frac{7 x}{- 1}$ .

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

Step 3.4.2.3.1.2

Rewrite $- 1 \cdot (7 x)$ as $- (7 x)$ .

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

Step 3.4.2.3.1.3

Multiply $7$ by $- 1$ .

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

Step 3.4.2.3.1.4

Divide $- 3$ by $- 1$ .

$y = - 7 x + 3$

Step 4

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

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

Step 5

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Cancel the common factor of $7$ .

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Step 5.2.3.1.1

Factor $7$ out of $- 7$ .

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

Step 5.2.3.1.2

Cancel the common factor.

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

Step 5.2.3.1.3

Rewrite the expression.

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

Step 5.2.3.2

Apply the distributive property.

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

Step 5.2.3.3

Multiply $- 1$ by $3$ .

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

Step 5.2.3.4

Multiply $- 1 (- x)$ .

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Step 5.2.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.3.4.2

Multiply $x$ by $1$ .

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

Step 5.2.4

Combine the opposite terms in $- 3 + x + 3$ .

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Step 5.2.4.1

Add $- 3$ and $3$ .

$f^{- 1} (\frac{3 - x}{7}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Multiply $- 7$ by $- 1$ .

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

Step 5.3.3.3

Multiply $- 1$ by $3$ .

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

Step 5.3.3.4

Subtract $3$ from $3$ .

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

Step 5.3.3.5

Add $7 x$ and $0$ .

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

Step 5.3.4

Cancel the common factor of $7$ .

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Step 5.3.4.1

Cancel the common factor.

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

Step 5.3.4.2

Divide $x$ by $1$ .

$f (- 7 x + 3) = x$

Step 5.4

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

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