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Understanding the concept of substituting variables is essential in the realm of mathematics, particularly when evaluating functions. The process involves replacing the function's variable with a specific value or expression to compute the function's output for that input.

Imagine the variable in a function as a placeholder or an empty seat. When we 'substitute' a variable, we are effectively placing a new value onto that seat. For example, if we have a function defined as \(f(x)\) and we want to evaluate this function at \(x = a\), we replace every instance of \(x\) with \(a\). Thus, \(f(a)\) represents the function's value at this particular point. This technique allows us to find the output of the function for any given input within its domain.

In the provided exercise, evaluating \(f(a)\), \(f(a+1)\), and \(f\left(\frac{1}{2}\right)\) requires us to substitute \(x\) with the given expressions and simplify the result. This substitution is critical in mathematics because it provides a direct method to compute the values of functions at specific points and is widely used in calculus, algebra, and more advanced fields like differential equations and linear algebra.

Arithmetic operations are the foundation of all mathematical computations. These operations include addition, subtraction, multiplication, and division. When evaluating functions, especially rational functions, it is inevitable to perform arithmetic operations to simplify expressions and reach the final result.

Following the substitution of a variable, we often have to simplify the resulting expression. This includes combining like terms, multiplying or dividing numbers, and executing any addition or subtraction as dictated by the expression. In the exercise, for instance, to evaluate \(f(a+1)\), we substitute \(a+1\) into the function and proceed with arithmetic operations like multiplying the \(2\) by \(a+1\) and then adding \(1\), leading to a simplified form.

Even simple arithmetic can sometimes lead to errors if we rush or overlook the order of operations—also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Thus, careful and systematic application of arithmetic operations ensures accuracy in evaluating functions.

Rational functions are ratios of two polynomial functions where the denominator is not allowed to be zero, as division by zero is undefined in mathematics. The generic form of a rational function is \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\).

Rational functions can model a wide range of real-world phenomena, from rates of change to optimization problems. They have characteristic features such as asymptotes, intercepts, and can have discontinuities, where the function is undefined.

In the context of our exercise, \(f(x) = \frac{1}{2x+1}\) is an example of a simple rational function. The function changes its value as \(x\) is substituted and often requires simplification after substitution. As shown in the solution, these functions must be handled with care to avoid undefined expressions, such as division by zero, which would occur if we substituted \(x = -\frac{1}{2}\) in this example. The elegance of rational functions lies in their complexity and ability to encapsulate intricate relationships, making them a fascinating subject in mathematics.