91Ó°ÊÓ

Understanding function notation is the first step when dealing with functions like those in the given exercise. Function notation uses symbols to represent a function's rule and indicates input and output clearly.
For example, given a function \( f(x) = -3x + 1 \), the symbol \( f \) names the function, and \( x \) represents the input variable.
When we see \( f(-1) \), it tells us to substitute -1 into the function \( f \). Essentially, \( f(x) \) converts to \( f(-1) \).
This notation is crucial because it clearly communicates which value to substitute into the function and ensures we're working with the correct function and corresponding values.
Functions are usually defined by an equation or rule, such as \( f(x) = -3x + 1 \) or \( g(x) = x^2 + 2 \). These notations help in easily identifying and applying different functions.

Substitution is a fundamental technique used to evaluate a function at a particular point. Let's take a closer look at how this process works.
First, identify the input value you need to substitute. From the exercise, we need to find \( f(-1) \). Substitution involves replacing the variable \( x \) with the given input value. So, for \( f(x) = -3x + 1 \), we substitute -1 for \( x \):

• \( f(-1) = -3(-1) + 1 \)

After substitution, simplify the expression to find the output value. In this case:

• \( f(-1) = 3 + 1 = 4 \)

Similarly, for another function like \( g(x) = x^2 + 2 \), substituting -1 for \( x \) gives:

• \( g(-1) = (-1)^2 + 2 = 1 + 2 = 3 \)

By substituting the input values correctly, we can determine the corresponding outputs for any given function.

After evaluating functions at specific points, another common operation is the addition of functions. This involves adding the output values of two functions evaluated at the same point.
In our example, we calculated the values as follows:

• \( f(-1) = 4 \)
• \( g(-1) = 3 \)

To find the sum of \( f(-1) \) and \( g(-1) \), simply add these results together:

• \( f(-1) + g(-1) = 4 + 3 = 7 \)

This method of adding the outputs of functions evaluated at the same point helps in combining the effects or results of different functions.
Remember, when performing function addition, ensure both functions are properly evaluated at the same input value to get an accurate result.