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The process of substitution is a fundamental mathematical technique used to find the output of a function when specific values are input into it. By replacing the variable in a function with a specific number or expression, we can directly calculate the corresponding function value.
In our exercise, to evaluate the function at different points, we substitute the values of the variable one by one.

• When evaluating at a point like \( a \), we substitute \( a \) for \( x \) in the function \( f(x) = \frac{1}{x+1} \) to get \( f(a) = \frac{1}{a+1} \).
• For \( a+1 \), the substitution changes the function to \( f(a+1) = \frac{1}{a+2} \).
• When the function is evaluated at \( \frac{1}{2} \), the substitution results in \( f\left(\frac{1}{2}\right) = \frac{2}{3} \).

Substitution is not only crucial for evaluating functions, but it also plays a significant role in solving equations and mathematical proofs. This method helps simplify expressions and allows us to solve complex problems by dealing with smaller, more manageable parts.

Rational functions are a type of mathematical function defined as the ratio of two polynomials. Think of them like fractions, but instead of simple numbers, they involve polynomial expressions in the numerator and the denominator.
The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \) to ensure the denominator isn't zero.
In our exercise, the function \( f(x) = \frac{1}{x+1} \) is a simple rational function.

• The numerator \( 1 \) is a constant polynomial.
• The denominator \( x+1 \) is a first-degree polynomial.

Rational functions are a central concept in algebra and are used in various practical applications, including modeling real-world phenomena. They have specific properties and rules, such as domain restrictions, due to divisions by zero.
Understanding rational functions helps in graphing them and solving complex equations that involve these types of expressions.

Algebraic manipulation involves rearranging and simplifying algebraic expressions to extract meaningful results or solutions. It's a set of techniques used throughout mathematics to make expressions easier to work with and understand.
In the context of our exercise, algebraic manipulation comes into play when substituting and simplifying the function values. For example, when evaluating \( f\left(\frac{1}{2}\right) \):

• Insert \( \frac{1}{2} \) into the denominator to get \( \frac{1}{\frac{1}{2}+1} \).
• Simplify the expression \( \frac{1}{\frac{3}{2}} \), which requires understanding """division of fractions""" to rewrite it as \( \frac{2}{3} \).

Algebraic manipulation is indispensable in simplifying expressions, solving equations, and performing higher-level calculus. Mastery of this skill is essential for more advanced mathematics and broader scientific applications, where complex expressions need simplifying for effective analysis.