Finite Math Examples

$f (t) = - 4 u (t - 4) + 4 u (t - 3)$

Step 1

Write $f (t) = - 4 u (t - 4) + 4 u (t - 3)$ as an equation.

$y = - 4 u (t - 4) + 4 u (t - 3)$

Step 2

Interchange the variables.

$u = - 4 y (t - 4) + 4 y (t - 3)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 4 y (t - 4) + 4 y (t - 3) = u$ .

$- 4 y (t - 4) + 4 y (t - 3) = u$

Step 3.2

Simplify $- 4 y (t - 4) + 4 y (t - 3)$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the distributive property.

$- 4 y t - 4 y \cdot - 4 + 4 y (t - 3) = u$

Step 3.2.1.2

Multiply $- 4$ by $- 4$ .

$- 4 y t + 16 y + 4 y (t - 3) = u$

Step 3.2.1.3

Apply the distributive property.

$- 4 y t + 16 y + 4 y t + 4 y \cdot - 3 = u$

Step 3.2.1.4

Multiply $- 3$ by $4$ .

$- 4 y t + 16 y + 4 y t - 12 y = u$

Step 3.2.2

Simplify by adding terms.

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Step 3.2.2.1

Combine the opposite terms in $- 4 y t + 16 y + 4 y t - 12 y$ .

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Step 3.2.2.1.1

Add $- 4 y t$ and $4 y t$ .

$16 y + 0 - 12 y = u$

Step 3.2.2.1.2

Add $16 y$ and $0$ .

$16 y - 12 y = u$

Step 3.2.2.2

Subtract $12 y$ from $16 y$ .

$4 y = u$

Step 3.3

Divide each term in $4 y = u$ by $4$ and simplify.

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Step 3.3.1

Divide each term in $4 y = u$ by $4$ .

$\frac{4 y}{4} = \frac{u}{4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{u}{4}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{u}{4}$

Step 4

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

$f^{- 1} (u) = \frac{u}{4}$

Step 5

Verify if $f^{- 1} (u) = \frac{u}{4}$ is the inverse of $f (u) = - 4 u (t - 4) + 4 u (t - 3)$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (- 4 u (t - 4) + 4 u (t - 3))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{- 4 u (t - 4) + 4 u (t - 3)}{4}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Factor $4 u$ out of $- 4 u (t - 4) + 4 u (t - 3)$ .

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Step 5.2.3.1.1

Factor $4 u$ out of $- 4 u (t - 4)$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u (- 1 (t - 4)) + 4 u (t - 3)}{4}$

Step 5.2.3.1.2

Factor $4 u$ out of $4 u (- 1 (t - 4)) + 4 u (t - 3)$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u (- 1 (t - 4) + t - 3)}{4}$

Step 5.2.3.2

Apply the distributive property.

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u (- 1 t - 1 \cdot - 4 + t - 3)}{4}$

Step 5.2.3.3

Rewrite $- 1 t$ as $- t$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u (- t - 1 \cdot - 4 + t - 3)}{4}$

Step 5.2.3.4

Multiply $- 1$ by $- 4$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u (- t + 4 + t - 3)}{4}$

Step 5.2.3.5

Add $- t$ and $t$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u (0 + 4 - 3)}{4}$

Step 5.2.3.6

Add $0$ and $4$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u (4 - 3)}{4}$

Step 5.2.3.7

Subtract $3$ from $4$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u \cdot 1}{4}$

Step 5.2.3.8

Multiply $4$ by $1$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u}{4}$

Step 5.2.4

Cancel the common factor of $4$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = \frac{4 u}{4}$

Step 5.2.4.2

Divide $u$ by $1$ .

$f^{- 1} (- 4 u (t - 4) + 4 u (t - 3)) = u$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{u}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{u}{4}) = - 4 \frac{u}{4} (t - 4) + 4 (\frac{u}{4}) (t - 3)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Cancel the common factor of $4$ .

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Step 5.3.3.1.1

Factor $4$ out of $- 4$ .

$f (\frac{u}{4}) = 4 (- 1) (\frac{u}{4}) \cdot (t - 4) + 4 (\frac{u}{4}) \cdot (t - 3)$

Step 5.3.3.1.2

Cancel the common factor.

$f (\frac{u}{4}) = 4 \cdot (- 1 \frac{u}{4}) \cdot (t - 4) + 4 (\frac{u}{4}) \cdot (t - 3)$

Step 5.3.3.1.3

Rewrite the expression.

$f (\frac{u}{4}) = - 1 u (t - 4) + 4 (\frac{u}{4}) \cdot (t - 3)$

Step 5.3.3.2

Rewrite $- 1 u$ as $- u$ .

$f (\frac{u}{4}) = - u (t - 4) + 4 (\frac{u}{4}) \cdot (t - 3)$

Step 5.3.3.3

Apply the distributive property.

$f (\frac{u}{4}) = - u t - u \cdot - 4 + 4 (\frac{u}{4}) \cdot (t - 3)$

Step 5.3.3.4

Multiply $- 4$ by $- 1$ .

$f (\frac{u}{4}) = - u t + 4 u + 4 (\frac{u}{4}) \cdot (t - 3)$

Step 5.3.3.5

Cancel the common factor of $4$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{u}{4}) = - u t + 4 u + 4 (\frac{u}{4}) \cdot (t - 3)$

Step 5.3.3.5.2

Rewrite the expression.

$f (\frac{u}{4}) = - u t + 4 u + u (t - 3)$

Step 5.3.3.6

Apply the distributive property.

$f (\frac{u}{4}) = - u t + 4 u + u t + u \cdot - 3$

Step 5.3.3.7

Move $- 3$ to the left of $u$ .

$f (\frac{u}{4}) = - u t + 4 u + u t - 3 u$

Step 5.3.4

Simplify by adding terms.

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Step 5.3.4.1

Combine the opposite terms in $- u t + 4 u + u t - 3 u$ .

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Step 5.3.4.1.1

Add $- u t$ and $u t$ .

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

Step 5.3.4.1.2

Add $4 u$ and $0$ .

$f (\frac{u}{4}) = 4 u - 3 u$

Step 5.3.4.2

Subtract $3 u$ from $4 u$ .

$f (\frac{u}{4}) = u$

Step 5.4

Since $f^{- 1} (f (u)) = u$ and $f (f^{- 1} (u)) = u$ , then $f^{- 1} (u) = \frac{u}{4}$ is the inverse of $f (u) = - 4 u (t - 4) + 4 u (t - 3)$ .

$f^{- 1} (u) = \frac{u}{4}$