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Evaluating a function means finding the value of the function at a particular value of the variable, often denoted as \( x \). In simple terms, you're plugging a number into a function to see what result you get out. For example, if you have a function like \( f(x) = x - 2 \), to find \( f(6) \) means you substitute \( 6 \) for \( x \) in the function, and calculate the result.

Evaluating functions can become more interesting when dealing with composite functions, where one function is plugged into another. This process involves multiple steps of evaluation and requires attention to detail.

• Start with the innermost function.
• Find its value and use this result as the input for the next function.
• Continue until all nested functions are evaluated.

A mathematical function is a relation between a set of inputs and a set of possible outputs. Each input is related to exactly one output. Think of it as a machine that takes an input, processes it according to a specific rule, and gives out an output. Common examples include linear functions, like \( f(x) = x + 2 \), and polynomial functions.

Functions have specific components and features:

• An expression defining the rule of the function such as \( x - 2 \).
• An input value, which is substituted into the rule.
• An output that results from the calculation.

Functions like \( g(x) = 3x \) are linear and illustrate how outputs change proportionally to the input.

Solving a problem using a step-by-step approach involves breaking down the problem into manageable parts. This makes complex problems easier to tackle, especially when dealing with compositions of functions. Here's how to use step-by-step solutions effectively:

First, identify each function involved. In our example, there are two functions: \( f(x) = x - 2 \) and \( g(x) = 3x \). Start by evaluating the innermost function.

In the exercise, you begin with \( g(6) \) by plugging in \( 6 \) into \( g(x) \):

• \( g(6) = 3 \times 6 = 18 \)

Next, you take this output and use it as the input for the next function, \( f(x) \):

• \( f(18) = 18 - 2 = 16 \)

By following this ordered process, each step leads naturally to the next, reducing errors and ensuring you get the right answer.