Model with rational coefficients

A reservoir contains 3/2 liter of water and receives 1/4 liter per minute. After t minutes, the volume is V(t) = 3/2 + t/4, while the described process continues. For t = 2, we have V(2) = 3/2 + 1/2 = 2 liters.

The domain depends on the situation: here t ≥ 0. Time can include fractions of a minute; we don’t need to restrict it to integers. If t is rational, V(t) will also be rational.

The denominator must exist and be non-zero

The expression f(x) = (x + 1)/(x − 2) is not defined in x = 2. If we study rational inputs, its domain is ℚ without the number 2. For x = 1/2, f(1/2) = (3/2)/(−3/2) = −1.

In the simplification (x² − 4)/(x − 2) = x + 2, we must keep x ≠ 2. Canceling factors allows for simpler writing, but does not recover a prohibited entry in the original expression.

Model challenge

In the reservoir, after how many minutes will the volume be 3 liters? In the expression f(x), what are the values ​​of f(0) and f(3)?

Check out the strategies

3/2 + t/4 = 3 implies t/4 = 3/2, therefore t = 6 minutes. Now f(0) = 1/(−2) = −1/2 and f(3) = 4/1 = 4. Entry 2 remains deleted.