Algebra Examples

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

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 $y$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

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

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Divide $q$ by $1$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

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

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Step 5.3.3.1

Cancel the common factor.

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

Step 5.3.3.2

Rewrite the expression.

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

Step 5.4

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

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