Computing Rational Functions

For the given functions and , do the following computation. Choose the best answer.

and

Find

Question

Computing Rational Functions

For the given functions  and , do the following computation. Choose the best answer.

and

Find

BlackTom AI · Accepted Answer

To start, note the problem involves dividing two rational expressions. We are given f(x) = (5x + 1)/(8x − 1) and g(x) = (3x)/(8x − 1). The task is to compute (f/g)(x) = f(x) / g(x).

First, set up the division of fractions: f(x)/g(x) = [(5x + 1)/(8x − 1)] ÷ [3x/(8x − 1)]. This is equivalent to multiplying by the reciprocal: [(5x + 1)/(8x − 1)] × [(8x − 1)/(3x)].

Next, observe a common factor cancellation: the (8x − 1) factor appears in both the numerator and denominator across the two fractions, so they cancel when multiplied. This simplifies the expression to (5x + 1)/(3x).

Now, consider the domain considerations briefly: as long as x ≠ 0 and 8x − 1 ≠ 0 (i.e., x ≠ 1/8), the simplification is valid. The stepwise cancellation is permissible because we are multiplying by the reciprocal and the original denominators are nonzero.

Looking at possible final forms one could write for the result, the expression (5x + 1)/(3x) is the simplified form after cancellation. Any other form would either reintroduce the canceled factors or would not be equivalent across the domain.

MAT-265-OD24-02-7799

Computing Rational Functions For the given functions and , do the following computation. Choose the best answer. and Find

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