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When you encounter the term "function substitution," it essentially involves replacing a variable within a function's expression with another value or expression. This process is fundamental in mathematics, especially for evaluating functions at specific points. To illustrate, take a function like \( h(z) = \frac{1}{z+7} \). If we want to evaluate it at \( z = -8 \), we substitute \( -8 \) for \( z \). This simple act of replacing \( z \) in the function expression with \( -8 \) is what we call substitution.

Substitution can also be used in more complex scenarios where instead of a single numeric value, a different expression might replace the variable. For instance, you could substitute \( z = 2x \) into \( h(z) \). Understanding this concept is critical as it is a foundational skill in algebra and calculus, allowing you to explore the behavior of functions under various transformations.

Rational functions are mathematical expressions defined as the ratios of two polynomials. They are written in the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomial expressions and \( q(x) eq 0 \). When examining the function \( h(z) = \frac{1}{z+7} \), you can see it fits the structure of a rational function because it's a ratio where the numerator is \( 1 \) (a constant polynomial) and the denominator is \( z + 7 \) (a linear polynomial).

Rational functions can have interesting properties such as vertical asymptotes, horizontal asymptotes, and potentially holes which can occur due to common factors in \( p(x) \) and \( q(x) \). In our specific case, the vertical asymptote happens where the denominator equals zero, so for \( h(z) \), the function is undefined at \( z = -7 \). As you advance in math, knowing these properties helps in sketching graphs and understanding the behavior of such functions.

Function evaluation is the process of finding the output of a function given a particular input. It involves two main steps: substitution and simplification. After substituting the given input into the function, you need to simplify the expression to obtain a numerical result.

In our example, \( h(z) = \frac{1}{z+7} \), you need to evaluate \( h(-8) \). The steps you follow are: first substitute \(-8\) for \(z\), which transforms the expression into \( \frac{1}{-8+7} \). Next, simplify the result of the operation \(-8+7\) which equals \(-1\). Thus, the expression \( \frac{1}{-1} \) simplifies to \(-1\).

Function evaluation is useful in many areas of mathematics and science because it lets you calculate specific outputs of a function, making these abstract expressions tangible and applicable in practical scenarios.