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Evaluating functions is a fundamental concept in mathematics. It involves finding the value of a function for a specific input. In our exercise, we're given a function, say \(G(x) = 5 - x\). To evaluate this function at a point (for instance, when \(x = 6\)), we substitute \(6\) into the function instead of \(x\) and then perform the arithmetic operations to find the value.

Here's the step-by-step process for evaluating \(G(6)\):

• Start with the expression: \(G(x) = 5 - x\)
• Substitute \(x = 6\): \(G(6) = 5 - 6\)
• Simplify the expression: \(5 - 6 = -1\)

This simple substitution allows us to evaluate the function at any given value, thereby helping us analyze its behavior and solve more complex problems.

Function multiplication is another key concept, often denoted as \((F \cdot G)(x)\). This means multiplying the results of two functions evaluated at the same point. In our example, the exercise asks us to find \((G \cdot G)(6)\).

Notice that this means we are multiplying \(G(6)\) by itself. Here's how to proceed:

• First, evaluate \(G(6)\) (which we already did and found to be \(-1\))
• Next, multiply the result by itself: \(G(6) \cdot G(6) = -1 \cdot -1 = 1\)

By performing these operations, function multiplication helps in combining the outputs of functions, yielding new results and insights into the behavior of the compositions and interactions between functions.

Understanding algebraic expressions is crucial as they form the backbone of evaluating and manipulating functions. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. For example, the function \(G(x) = 5 - x\) is an algebraic expression.

To work with algebraic expressions, follow these steps:

• Identify the variables and constants involved
• Follow the proper order of operations (PEMDAS - Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right))
• Substitute the given values into the expression and simplify

Applying these principles, we simplify and solve expressions step-by-step. This ensures accuracy and provides a clear path to the solution. In our exercise, the algebraic manipulation involved substituting \(x = 6\) into \(G(x) = 5 - x\) and then performing the subtraction to find \(-1\). By consistently practicing these steps, you’ll find solving algebraic expressions increasingly intuitive and straightforward.