Explain why rational expressions often have non-permissible values for some variables (i.e.,
restrictions on the variables). Give examples.
Rational expressions often have non-permissible values for some variables because they involve division by variables, and division by zero is undefined in mathematics. These non-permissible values are also referred to as restrictions on the variables.
For example, consider the rational expression:
f(x) = (x + 3) / (x - 2)
In this expression, there is a restriction on the variable x. The denominator (x - 2) cannot be equal to zero, as division by zero is undefined. Therefore, x - 2 ≠ 0, which gives us the restriction x ≠ 2.
If x were equal to 2, then the expression would involve dividing by zero, leading to an undefined result. So, the non-permissible value in this case is 2.
Another example is the rational expression:
g(t) = √(t + 5) / (t - 4)
In this case, the expression involves taking the square root of (t + 5). It is important to note that the square root of a negative number is undefined in the real number system. So, t + 5 ≥ 0 for the expression to be defined.
Therefore, the restriction in this case is t ≥ -5.
These examples demonstrate that rational expressions often have non-permissible values or restrictions on the variables to ensure that the division operation is well-defined and avoids undefined results.