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Evaluating functions means finding the output of a function for a given input value. In mathematical terms, if you have a function \(f(x)\), evaluating it at \(x\) means you are calculating \(f(x)\). For instance, if \(f(x) = 2x + 3\) and you want to evaluate this function at \(x = 4\), you would plug in 4 for \(x\) and get \(f(4) = 2(4) + 3 = 11\).

When presented with a table of function values, as in the exercise, you can see directly what \(f(x)\) equals for specific \(x\) values. Similarly, another table showing \(g(x)\) gives the respective output values of the function \(g\) for different inputs. Understanding this concept is crucial because it helps you verify whether functions are inverses of each other by comparing output values.

Function composition involves creating a new function by applying one function to the result of another. If you have two functions \(f\) and \(g\), their composition \((f \, \text{circ} \, g)(x)\) is represented mathematically as \(f(g(x))\). This means you first apply \(g\) to \(x\) and then apply \(f\) to the result of \(g(x)\).

To check if two functions are inverses of each other, we look at their compositions. A key property of inverse functions \(f\) and \(g\) is that \(f(g(x)) = x\) and \(g(f(x)) = x\). This means if you start with an input \(x\), apply \(g\) to it, and then apply \(f\) to the result, you should get the original input \(x\).

In the given exercise:

• You first take the value of \(x\) from the second table to find \(g(x)\).
• Then, use this \(g(x)\) value as the input for the first function \(f\).
• If \(f(g(x)) = x\) holds true for all values, then the functions are inverses.

However, from the step-by-step solution, we see this verification fails, proving that \(f\) and \(g\) are not inverse functions.

Precalculus functions bridge the gap between algebra and calculus by introducing important concepts and methods for describing mathematical phenomena. Understanding how to work with various functions is a key part of precalculus.

Some key ideas include:

• Domain and Range: The domain is the set of all possible inputs for a function, while the range is the set of all possible outputs.
• Inverse Functions: As mentioned earlier, two functions are inverses if each one's output can serve as the input to the other and return the original input value.
• Evaluating and Composing Functions: These skills help students understand how functions can be manipulated and used together to form new functions.
• Tables and Graphs: Presenting function values in tables or graphs helps visualise the function's behaviour.

Mastering these elements is essential before moving on to more advanced topics in calculus, as they provide the foundation for understanding limits, derivatives, and integrals.