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Calculus Examples

f (x) = 6 - 4 x

$f (x) = 6 - 4 x$

Step 1

Write

f (x) = 6 - 4 x

$f (x) = 6 - 4 x$ as an equation.

y = 6 - 4 x

$y = 6 - 4 x$

Step 2

Interchange the variables.

x = 6 - 4 y

$x = 6 - 4 y$

Step 3

Solve for

y

$y$ .

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Step 3.1

Rewrite the equation as

6 - 4 y = x

$6 - 4 y = x$ .

6 - 4 y = x

$6 - 4 y = x$

Step 3.2

Subtract

6

$6$ from both sides of the equation.

- 4 y = x - 6

$- 4 y = x - 6$

Step 3.3

Divide each term in

- 4 y = x - 6

$- 4 y = x - 6$ by

- 4

$- 4$ and simplify.

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Step 3.3.1

Divide each term in

- 4 y = x - 6

$- 4 y = x - 6$ by

- 4

$- 4$ .

\frac{- 4 y}{- 4} = \frac{x}{- 4} + \frac{- 6}{- 4}

$\frac{- 4 y}{- 4} = \frac{x}{- 4} + \frac{- 6}{- 4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of

- 4

$- 4$ .

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Step 3.3.2.1.1

Cancel the common factor.

\frac{- 4 y}{- 4} = \frac{x}{- 4} + \frac{- 6}{- 4}

$\frac{- 4 y}{- 4} = \frac{x}{- 4} + \frac{- 6}{- 4}$

Step 3.3.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{x}{- 4} + \frac{- 6}{- 4}

$y = \frac{x}{- 4} + \frac{- 6}{- 4}$

y = \frac{x}{- 4} + \frac{- 6}{- 4}

$y = \frac{x}{- 4} + \frac{- 6}{- 4}$

y = \frac{x}{- 4} + \frac{- 6}{- 4}

$y = \frac{x}{- 4} + \frac{- 6}{- 4}$

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 in front of the fraction.

y = - \frac{x}{4} + \frac{- 6}{- 4}

$y = - \frac{x}{4} + \frac{- 6}{- 4}$

Step 3.3.3.1.2

Cancel the common factor of

- 6

$- 6$ and

- 4

$- 4$ .

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Step 3.3.3.1.2.1

Factor

- 2

$- 2$ out of

- 6

$- 6$ .

y = - \frac{x}{4} + \frac{- 2 (3)}{- 4}

$y = - \frac{x}{4} + \frac{- 2 (3)}{- 4}$

Step 3.3.3.1.2.2

Cancel the common factors.

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Step 3.3.3.1.2.2.1

Factor

- 2

$- 2$ out of

- 4

$- 4$ .

y = - \frac{x}{4} + \frac{- 2 \cdot 3}{- 2 \cdot 2}

$y = - \frac{x}{4} + \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Step 3.3.3.1.2.2.2

Cancel the common factor.

y = - \frac{x}{4} + \frac{- 2 \cdot 3}{- 2 \cdot 2}

$y = - \frac{x}{4} + \frac{- 2 \cdot 3}{- 2 \cdot 2}$

Step 3.3.3.1.2.2.3

Rewrite the expression.

y = - \frac{x}{4} + \frac{3}{2}

$y = - \frac{x}{4} + \frac{3}{2}$

y = - \frac{x}{4} + \frac{3}{2}

$y = - \frac{x}{4} + \frac{3}{2}$

y = - \frac{x}{4} + \frac{3}{2}

$y = - \frac{x}{4} + \frac{3}{2}$

y = - \frac{x}{4} + \frac{3}{2}

$y = - \frac{x}{4} + \frac{3}{2}$

y = - \frac{x}{4} + \frac{3}{2}

$y = - \frac{x}{4} + \frac{3}{2}$

y = - \frac{x}{4} + \frac{3}{2}

$y = - \frac{x}{4} + \frac{3}{2}$

y = - \frac{x}{4} + \frac{3}{2}

$y = - \frac{x}{4} + \frac{3}{2}$

Step 4

Replace

y

$y$ with

f^{- 1} (x)

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

f^{- 1} (x) = - \frac{x}{4} + \frac{3}{2}

$f^{- 1} (x) = - \frac{x}{4} + \frac{3}{2}$

Step 5

Verify if

f^{- 1} (x) = - \frac{x}{4} + \frac{3}{2}

$f^{- 1} (x) = - \frac{x}{4} + \frac{3}{2}$ is the inverse of

f (x) = 6 - 4 x

$f (x) = 6 - 4 x$ .

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Step 5.1

To verify the inverse, check if

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

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

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

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

Step 5.2

Evaluate

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

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

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Step 5.2.1

Set up the composite result function.

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

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

Step 5.2.2

Evaluate

f^{- 1} (6 - 4 x)

$f^{- 1} (6 - 4 x)$ by substituting in the value of

f

$f$ into

f^{- 1}

$f^{- 1}$ .

f^{- 1} (6 - 4 x) = - \frac{6 - 4 x}{4} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{6 - 4 x}{4} + \frac{3}{2}$

Step 5.2.3

Simplify terms.

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Step 5.2.3.1

Cancel the common factor of

6 - 4 x

$6 - 4 x$ and

4

$4$ .

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Step 5.2.3.1.1

Factor

2

$2$ out of

6

$6$ .

f^{- 1} (6 - 4 x) = - \frac{2 (3) - 4 x}{4} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{2 (3) - 4 x}{4} + \frac{3}{2}$

Step 5.2.3.1.2

Factor

2

$2$ out of

- 4 x

$- 4 x$ .

f^{- 1} (6 - 4 x) = - \frac{2 (3) + 2 (- 2 x)}{4} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{2 (3) + 2 (- 2 x)}{4} + \frac{3}{2}$

Step 5.2.3.1.3

Factor

2

$2$ out of

2 (3) + 2 (- 2 x)

$2 (3) + 2 (- 2 x)$ .

f^{- 1} (6 - 4 x) = - \frac{2 (3 - 2 x)}{4} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{2 (3 - 2 x)}{4} + \frac{3}{2}$

Step 5.2.3.1.4

Cancel the common factors.

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Step 5.2.3.1.4.1

Factor

2

$2$ out of

4

$4$ .

f^{- 1} (6 - 4 x) = - \frac{2 (3 - 2 x)}{2 \cdot 2} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{2 (3 - 2 x)}{2 \cdot 2} + \frac{3}{2}$

Step 5.2.3.1.4.2

Cancel the common factor.

f^{- 1} (6 - 4 x) = - \frac{2 (3 - 2 x)}{2 \cdot 2} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{2 (3 - 2 x)}{2 \cdot 2} + \frac{3}{2}$

Step 5.2.3.1.4.3

Rewrite the expression.

f^{- 1} (6 - 4 x) = - \frac{3 - 2 x}{2} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{3 - 2 x}{2} + \frac{3}{2}$

f^{- 1} (6 - 4 x) = - \frac{3 - 2 x}{2} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{3 - 2 x}{2} + \frac{3}{2}$

f^{- 1} (6 - 4 x) = - \frac{3 - 2 x}{2} + \frac{3}{2}

$f^{- 1} (6 - 4 x) = - \frac{3 - 2 x}{2} + \frac{3}{2}$

Step 5.2.3.2

Combine the numerators over the common denominator.

f^{- 1} (6 - 4 x) = \frac{- (3 - 2 x) + 3}{2}

$f^{- 1} (6 - 4 x) = \frac{- (3 - 2 x) + 3}{2}$

f^{- 1} (6 - 4 x) = \frac{- (3 - 2 x) + 3}{2}

$f^{- 1} (6 - 4 x) = \frac{- (3 - 2 x) + 3}{2}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

f^{- 1} (6 - 4 x) = \frac{- 1 \cdot 3 - (- 2 x) + 3}{2}

$f^{- 1} (6 - 4 x) = \frac{- 1 \cdot 3 - (- 2 x) + 3}{2}$

Step 5.2.4.2

Multiply

- 1

$- 1$ by

3

$3$ .

f^{- 1} (6 - 4 x) = \frac{- 3 - (- 2 x) + 3}{2}

$f^{- 1} (6 - 4 x) = \frac{- 3 - (- 2 x) + 3}{2}$

Step 5.2.4.3

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

f^{- 1} (6 - 4 x) = \frac{- 3 + 2 x + 3}{2}

$f^{- 1} (6 - 4 x) = \frac{- 3 + 2 x + 3}{2}$

f^{- 1} (6 - 4 x) = \frac{- 3 + 2 x + 3}{2}

$f^{- 1} (6 - 4 x) = \frac{- 3 + 2 x + 3}{2}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in

- 3 + 2 x + 3

$- 3 + 2 x + 3$ .

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Step 5.2.5.1.1

Add

- 3

$- 3$ and

3

$3$ .

f^{- 1} (6 - 4 x) = \frac{2 x + 0}{2}

$f^{- 1} (6 - 4 x) = \frac{2 x + 0}{2}$

Step 5.2.5.1.2

Add

2 x

$2 x$ and

0

$0$ .

f^{- 1} (6 - 4 x) = \frac{2 x}{2}

$f^{- 1} (6 - 4 x) = \frac{2 x}{2}$

f^{- 1} (6 - 4 x) = \frac{2 x}{2}

$f^{- 1} (6 - 4 x) = \frac{2 x}{2}$

Step 5.2.5.2

Cancel the common factor of

2

$2$ .

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Step 5.2.5.2.1

Cancel the common factor.

f^{- 1} (6 - 4 x) = \frac{2 x}{2}

$f^{- 1} (6 - 4 x) = \frac{2 x}{2}$

Step 5.2.5.2.2

Divide

x

$x$ by

1

$1$ .

f^{- 1} (6 - 4 x) = x

$f^{- 1} (6 - 4 x) = x$

f^{- 1} (6 - 4 x) = x

$f^{- 1} (6 - 4 x) = x$

f^{- 1} (6 - 4 x) = x

$f^{- 1} (6 - 4 x) = x$

f^{- 1} (6 - 4 x) = x

$f^{- 1} (6 - 4 x) = x$

Step 5.3

Evaluate

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

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

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Step 5.3.1

Set up the composite result function.

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

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

Step 5.3.2

Evaluate

f (- \frac{x}{4} + \frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2})$ by substituting in the value of

f^{- 1}

$f^{- 1}$ into

f

$f$ .

f (- \frac{x}{4} + \frac{3}{2}) = 6 - 4 (- \frac{x}{4} + \frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 - 4 (- \frac{x}{4} + \frac{3}{2})$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

f (- \frac{x}{4} + \frac{3}{2}) = 6 - 4 (- \frac{x}{4}) - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 - 4 (- \frac{x}{4}) - 4 (\frac{3}{2})$

Step 5.3.3.2

Cancel the common factor of

4

$4$ .

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Step 5.3.3.2.1

Move the leading negative in

- \frac{x}{4}

$- \frac{x}{4}$ into the numerator.

f (- \frac{x}{4} + \frac{3}{2}) = 6 - 4 \frac{- x}{4} - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 - 4 \frac{- x}{4} - 4 (\frac{3}{2})$

Step 5.3.3.2.2

Factor

4

$4$ out of

- 4

$- 4$ .

f (- \frac{x}{4} + \frac{3}{2}) = 6 + 4 (- 1) (\frac{- x}{4}) - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + 4 (- 1) (\frac{- x}{4}) - 4 (\frac{3}{2})$

Step 5.3.3.2.3

Cancel the common factor.

f (- \frac{x}{4} + \frac{3}{2}) = 6 + 4 \cdot (- 1 \frac{- x}{4}) - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + 4 \cdot (- 1 \frac{- x}{4}) - 4 (\frac{3}{2})$

Step 5.3.3.2.4

Rewrite the expression.

f (- \frac{x}{4} + \frac{3}{2}) = 6 - 1 (- x) - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 - 1 (- x) - 4 (\frac{3}{2})$

f (- \frac{x}{4} + \frac{3}{2}) = 6 - 1 (- x) - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 - 1 (- x) - 4 (\frac{3}{2})$

Step 5.3.3.3

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

f (- \frac{x}{4} + \frac{3}{2}) = 6 + 1 x - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + 1 x - 4 (\frac{3}{2})$

Step 5.3.3.4

Multiply

x

$x$ by

1

$1$ .

f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 4 (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 4 (\frac{3}{2})$

Step 5.3.3.5

Cancel the common factor of

2

$2$ .

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Step 5.3.3.5.1

Factor

2

$2$ out of

- 4

$- 4$ .

f (- \frac{x}{4} + \frac{3}{2}) = 6 + x + 2 (- 2) (\frac{3}{2})

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + x + 2 (- 2) (\frac{3}{2})$

Step 5.3.3.5.2

Cancel the common factor.

f (- \frac{x}{4} + \frac{3}{2}) = 6 + x + 2 \cdot (- 2 (\frac{3}{2}))

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + x + 2 \cdot (- 2 (\frac{3}{2}))$

Step 5.3.3.5.3

Rewrite the expression.

f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 2 \cdot 3

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 2 \cdot 3$

f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 2 \cdot 3

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 2 \cdot 3$

Step 5.3.3.6

Multiply

- 2

$- 2$ by

3

$3$ .

f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 6

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 6$

f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 6

$f (- \frac{x}{4} + \frac{3}{2}) = 6 + x - 6$

Step 5.3.4

Combine the opposite terms in

$6 + x - 6$ .

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Step 5.3.4.1

Subtract

$6$ from

$6$ .

$f (- \frac{x}{4} + \frac{3}{2}) = x + 0$

Step 5.3.4.2

Add

$x$ and

$0$ .

$f (- \frac{x}{4} + \frac{3}{2}) = x$

$f (- \frac{x}{4} + \frac{3}{2}) = x$

$f (- \frac{x}{4} + \frac{3}{2}) = x$

Step 5.4

Since

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

$f (f^{- 1} (x)) = x$ , then

$f^{- 1} (x) = - \frac{x}{4} + \frac{3}{2}$ is the inverse of

$f (x) = 6 - 4 x$ .

$f^{- 1} (x) = - \frac{x}{4} + \frac{3}{2}$

$f^{- 1} (x) = - \frac{x}{4} + \frac{3}{2}$

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