Find the Inverse q(t)=q(1-e^(-t/a))

$q (t) = q (1 - e^{- \frac{t}{a}})$

Step 1

Write $q (t) = q (1 - e^{- \frac{t}{a}})$ as an equation.

$y = q (1 - e^{- \frac{t}{a}})$

Step 2

Interchange the variables.

$q = y (1 - e^{- \frac{t}{a}})$

Step 3

Solve for $y$ .

Rewrite the equation as $y (1 - e^{- \frac{t}{a}}) = q$ .

$y (1 - e^{- \frac{t}{a}}) = q$

Divide each term in $y (1 - e^{- \frac{t}{a}}) = q$ by $1 - e^{- \frac{t}{a}}$ and simplify.

Divide each term in $y (1 - e^{- \frac{t}{a}}) = q$ by $1 - e^{- \frac{t}{a}}$ .

$\frac{y (1 - e^{- \frac{t}{a}})}{1 - e^{- \frac{t}{a}}} = \frac{q}{1 - e^{- \frac{t}{a}}}$

Simplify the left side.

Cancel the common factor of $1 - e^{- \frac{t}{a}}$ .

Cancel the common factor.

$\frac{y (1 - e^{- \frac{t}{a}})}{1 - e^{- \frac{t}{a}}} = \frac{q}{1 - e^{- \frac{t}{a}}}$

Divide $y$ by $1$ .

$y = \frac{q}{1 - e^{- \frac{t}{a}}}$

Step 4

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

$q^{- 1} (q) = \frac{q}{1 - e^{- \frac{t}{a}}}$

Step 5

Verify if $q^{- 1} (q) = \frac{q}{1 - e^{- \frac{t}{a}}}$ is the inverse of $q (q) = q (1 - e^{- \frac{t}{a}})$ .

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

Evaluate $q^{- 1} (q (q))$ .

Set up the composite result function.

$q^{- 1} (q (q))$

Evaluate $q^{- 1} (q (1 - e^{- \frac{t}{a}}))$ by substituting in the value of $q$ into $q^{- 1}$ .

$q^{- 1} (q (1 - e^{- \frac{t}{a}})) = \frac{q (1 - e^{- \frac{t}{a}})}{1 - e^{- \frac{t}{a}}}$

Cancel the common factor of $1 - e^{- \frac{t}{a}}$ .

Cancel the common factor.

$q^{- 1} (q (1 - e^{- \frac{t}{a}})) = \frac{q (1 - e^{- \frac{t}{a}})}{1 - e^{- \frac{t}{a}}}$

Divide $q$ by $1$ .

$q^{- 1} (q (1 - e^{- \frac{t}{a}})) = q$

Evaluate $q (q^{- 1} (q))$ .

Set up the composite result function.

$q (q^{- 1} (q))$

Evaluate $q (\frac{q}{1 - e^{- \frac{t}{a}}})$ by substituting in the value of $q^{- 1}$ into $q$ .

$q (\frac{q}{1 - e^{- \frac{t}{a}}}) = (\frac{q}{1 - e^{- \frac{t}{a}}}) (1 - e^{- \frac{t}{a}})$

Cancel the common factor of $1 - e^{- \frac{t}{a}}$ .

Cancel the common factor.

$q (\frac{q}{1 - e^{- \frac{t}{a}}}) = \frac{q}{1 - e^{- \frac{t}{a}}} \cdot (1 - e^{- \frac{t}{a}})$

Rewrite the expression.

$q (\frac{q}{1 - e^{- \frac{t}{a}}}) = q$

Since $q^{- 1} (q (q)) = q$ and $q (q^{- 1} (q)) = q$ , then $q^{- 1} (q) = \frac{q}{1 - e^{- \frac{t}{a}}}$ is the inverse of $q (q) = q (1 - e^{- \frac{t}{a}})$ .

$q^{- 1} (q) = \frac{q}{1 - e^{- \frac{t}{a}}}$

Find the Inverse q(t)=q(1-e^(-t/a))

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