91Ó°ÊÓ

Rational functions are functions represented as the quotient of two polynomials. Quite simply, they are fractions where both the numerator and the denominator are polynomials.
These functions may have one or more terms in either the numerator or the denominator. For example, the function given in the exercise is:

\(\frac{8x}{(x-1)(x-2)}\).

Because the function involves a division, special care must be taken with the denominator: it should never be zero, since division by zero is undefined. This brings us to the concept of excluded values.

Excluded values are specific inputs (values of x) that make the denominator of a rational function equal to zero, hence rendering the function undefined.
For instance, in the function \(\frac{8x}{(x-1)(x-2)}\), we must find the values of \(x\) that cause the denominator \((x-1)(x-2)\) to equal zero.
We solve \((x-1)(x-2)=0\) to find which values need to be excluded from the domain. Solving this gives us x=1 and x=2, meaning these values are excluded.

Therefore, the domain does not include \(x=1\) and \(x=2\) since at these points, the function cannot be evaluated.

For a rational function to be valid, the denominator must be non-zero. This restriction is crucial because any input that zeroes out the denominator makes the function undefined. The steps to determine restrictions involve:

• Identifying the expression in the denominator.
• Setting the expression equal to zero.
• Solving for \(x\).

In the exercise, the expression in the denominator is \((x-1)(x-2)\). We set it to zero:
\((x-1)(x-2)=0\) and solve for x, giving x=1 and x=2.
These values are restricted because they make the denominator zero, which leads us to the final concept.

The domain of rational functions generally includes all real numbers except the excluded ones.
Real numbers encompass all the numbers on the number line: positive, negative, and zero. When determining the domain of the given function \(\frac{8x}{(x-1)(x-2)}\), we exclude only 1 and 2 from this set.

Thus, the domain of \(\frac{8x}{(x-1)(x-2)}\) is all real numbers except where the denominator is zero. That is all x such that x≠1 and x≠2. Summarized, the domain can be written as:
\(\text{Domain} = \(R - \{1, 2\}\)\)
Here, \(R\) represents the set of all real numbers, while the notation \(\{1, 2\}\) signifies the excluded values. Knowing this ensures we properly manage rational functions.