Algebra Examples

$f (x) = - x + 1$

Step 1

Write $f (x) = - x + 1$ as an equation.

$y = - x + 1$

Step 2

Interchange the variables.

$x = - y + 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- y + 1 = x$ .

$- y + 1 = x$

Step 3.2

Subtract $1$ from both sides of the equation.

$- y = x - 1$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

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

Step 3.3.2.2

Divide $y$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

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

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

Step 3.3.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

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

Step 3.3.3.1.3

Divide $- 1$ by $- 1$ .

$y = - x + 1$

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the distributive property.

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

Step 5.2.3.2

Multiply $- - x$ .

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Step 5.2.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.3.2.2

Multiply $x$ by $1$ .

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

Step 5.2.3.3

Multiply $- 1$ by $1$ .

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

Step 5.2.4

Combine the opposite terms in $x - 1 + 1$ .

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Step 5.2.4.1

Add $- 1$ and $1$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

$f (- x + 1) = - (- x + 1) + 1$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (- x + 1) = x - 1 \cdot 1 + 1$

Step 5.3.3.2

Multiply $- - x$ .

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Step 5.3.3.2.1

Multiply $- 1$ by $- 1$ .

$f (- x + 1) = 1 x - 1 \cdot 1 + 1$

Step 5.3.3.2.2

Multiply $x$ by $1$ .

$f (- x + 1) = x - 1 \cdot 1 + 1$

Step 5.3.3.3

Multiply $- 1$ by $1$ .

$f (- x + 1) = x - 1 + 1$

Step 5.3.4

Combine the opposite terms in $x - 1 + 1$ .

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Step 5.3.4.1

Add $- 1$ and $1$ .

$f (- x + 1) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (- x + 1) = x$

Step 5.4

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

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