Find the Inverse C(t)=4t-160

$C (t) = 4 t - 160$

Step 1

Write $C (t) = 4 t - 160$ as an equation.

$y = 4 t - 160$

Step 2

Interchange the variables.

$t = 4 y - 160$

Step 3

Solve for $y$ .

Rewrite the equation as $4 y - 160 = t$ .

$4 y - 160 = t$

Add $160$ to both sides of the equation.

$4 y = t + 160$

Divide each term in $4 y = t + 160$ by $4$ and simplify.

Divide each term in $4 y = t + 160$ by $4$ .

$\frac{4 y}{4} = \frac{t}{4} + \frac{160}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{t}{4} + \frac{160}{4}$

Divide $y$ by $1$ .

$y = \frac{t}{4} + \frac{160}{4}$

Simplify the right side.

Divide $160$ by $4$ .

$y = \frac{t}{4} + 40$

Step 4

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

$C^{- 1} (t) = \frac{t}{4} + 40$

Step 5

Verify if $C^{- 1} (t) = \frac{t}{4} + 40$ is the inverse of $C (t) = 4 t - 160$ .

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

Evaluate $C^{- 1} (C (t))$ .

Set up the composite result function.

$C^{- 1} (C (t))$

Evaluate $C^{- 1} (4 t - 160)$ by substituting in the value of $C$ into $C^{- 1}$ .

$C^{- 1} (4 t - 160) = \frac{4 t - 160}{4} + 40$

Cancel the common factor of $4 t - 160$ and $4$ .

Factor $4$ out of $4 t$ .

$C^{- 1} (4 t - 160) = \frac{4 (t) - 160}{4} + 40$

Factor $4$ out of $- 160$ .

$C^{- 1} (4 t - 160) = \frac{4 (t) + 4 \cdot - 40}{4} + 40$

Factor $4$ out of $4 (t) + 4 (- 40)$ .

$C^{- 1} (4 t - 160) = \frac{4 (t - 40)}{4} + 40$

Cancel the common factors.

Factor $4$ out of $4$ .

$C^{- 1} (4 t - 160) = \frac{4 (t - 40)}{4 (1)} + 40$

Cancel the common factor.

$C^{- 1} (4 t - 160) = \frac{4 (t - 40)}{4 \cdot 1} + 40$

Rewrite the expression.

$C^{- 1} (4 t - 160) = \frac{t - 40}{1} + 40$

Divide $t - 40$ by $1$ .

$C^{- 1} (4 t - 160) = t - 40 + 40$

Combine the opposite terms in $t - 40 + 40$ .

Add $- 40$ and $40$ .

$C^{- 1} (4 t - 160) = t + 0$

Add $t$ and $0$ .

$C^{- 1} (4 t - 160) = t$

Evaluate $C (C^{- 1} (t))$ .

Set up the composite result function.

$C (C^{- 1} (t))$

Evaluate $C (\frac{t}{4} + 40)$ by substituting in the value of $C^{- 1}$ into $C$ .

$C (\frac{t}{4} + 40) = 4 (\frac{t}{4} + 40) - 160$

Simplify each term.

Apply the distributive property.

$C (\frac{t}{4} + 40) = 4 (\frac{t}{4}) + 4 \cdot 40 - 160$

Cancel the common factor of $4$ .

Cancel the common factor.

$C (\frac{t}{4} + 40) = 4 (\frac{t}{4}) + 4 \cdot 40 - 160$

Rewrite the expression.

$C (\frac{t}{4} + 40) = t + 4 \cdot 40 - 160$

Multiply $4$ by $40$ .

$C (\frac{t}{4} + 40) = t + 160 - 160$

Combine the opposite terms in $t + 160 - 160$ .

Subtract $160$ from $160$ .

$C (\frac{t}{4} + 40) = t + 0$

Add $t$ and $0$ .

$C (\frac{t}{4} + 40) = t$

Since $C^{- 1} (C (t)) = t$ and $C (C^{- 1} (t)) = t$ , then $C^{- 1} (t) = \frac{t}{4} + 40$ is the inverse of $C (t) = 4 t - 160$ .

$C^{- 1} (t) = \frac{t}{4} + 40$

Find the Inverse C(t)=4t-160

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