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Rational functions are a type of mathematical expression, which consists of a ratio between two polynomial functions. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). This ensures that division by zero is avoided, which would otherwise make the function undefined.
In our exercise, the rational function is given by \( f(x) = \frac{2x}{x-1} \). Here, the numerator is the polynomial \( 2x \), and the denominator is \( x-1 \).
Some important characteristics of rational functions include:

• The domain, which consists of all real numbers except those that make the denominator zero, such as when \( x = 1 \) in this case.
• Rational functions can have vertical asymptotes, which occur at values that make the denominator zero.
• They may also have horizontal or oblique asymptotes based on the degrees of the polynomials in the numerator and denominator.
• Graphically, rational functions can take a variety of shapes, often having areas where they approach but never touch the asymptotes.

Understanding rational functions is key as they are common in calculus and emerge in various applied contexts.

Function evaluation is the process of calculating the output of a function for a given input. It involves substituting the input value into the function and performing the arithmetic operations indicated by the function's expression.
In our exercise, we first evaluate \( f(a) \) by substituting \( x = a \) into the rational function:

• Substitute \( a \) for \( x \) to get \( f(a) = \frac{2a}{a-1} \).

Next, we evaluate \( f(a+h) \) by substituting \( x = a+h \):

• Substitute \( a+h \) for \( x \) to obtain \( f(a+h) = \frac{2(a+h)}{a+h-1} \).

Function evaluation is a fundamental concept because it allows us to analyze how changes in the input affect the output of a function, which is crucial in both algebra and calculus.
Mastering function evaluation is essential for dealing with more complex concepts such as limit, continuity, and differentiability.

Algebraic simplification involves rewriting expressions in a simpler or more efficient form without changing their value. This skill is essential for solving equations, especially when dealing with rational functions and difference quotients.
In the provided solution, the goal is to simplify the difference quotient \( \frac{f(a+h) - f(a)}{h} \). We started by finding a common denominator for the terms in the numerator:

• The expression \( \frac{2(a+h)(a-1) - 2a(a+h-1)}{(a+h-1)(a-1)} \) involves expanding and simplifying polynomials.

The action plan in the process includes:

• Expanding each polynomial term: multiplying through the brackets.
• Combining like terms: zeroing out terms that cancel each other out.
• After simplification, the resulting numerator becomes \( -2h \), which is much simpler and straightforward.

Finally, dividing the entire expression by \( h \) leads to the simplified form \( \frac{-2}{(a+h-1)(a-1)} \).
Algebraic simplification reduces complexity, making expressions easier to work with and interpret. It is especially beneficial in calculus when evaluating limits and analyzing functions.