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Function notation is a way to name functions in a clear and concise manner. Instead of just saying a formula, function notation uses a name, like \(f\), and an input, like \(x\). For example, we write \(f(x) = -3x + 4\). This tells us that \(f(x)\) is a function where each input \(x\) is multiplied by \(-3\) and then 4 is added.
Function notation helps us understand what value we should substitute in and how the output changes when the input changes.
Let's dive a bit deeper:

• The name of the function (in this case, \(f\)) can vary. You might see \(g(x)\), \(h(x)\), etc.
• The input (inside the parentheses, here \(x\)) can be any number.
• The formula on the right side of the equation shows us exactly how to calculate the output from the input.

Simply put, function notation makes it easier to work with functions and see their relationships clearly.

The substitution method is a common technique used to evaluate functions. Here’s how it works:
The first step is to recognize the function we are working with. For example, our function is \( f(x) = -3x + 4 \).
Next, we need to substitute the given value of \(x\) into the function. In this exercise, we're asked to find \(f(0)\). So, we replace every \(x\) in the function with 0. This gives us \(-3(0) + 4\).
Finally, we perform the arithmetic to simplify and find the result.
To summarize:

• Identify the function.
• Substitute the given value for \(x\).
• Perform the calculation to find the output.

This method ensures that we correctly follow the rules defined by the function's formula.

Linear functions are a type of function where the relationship between the input and output is a straight line when graphed. They have the general form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
For the function in our exercise, \(f(x) = -3x + 4\), the slope \(m\) is \(-3\) and the y-intercept \(b\) is 4. This means that for every 1 unit increase in \(x\), the value of \(f(x)\) decreases by 3 units.
Characteristics of linear functions include:

• A constant rate of change; the slope (\(m\)) is consistent.
• The graph is a straight line.
• No exponent greater than 1 on the variable \(x\).

Linear functions are simpler to work with because you only need to deal with addition, subtraction, and multiplication by constants. They are fundamental in algebra and provide a foundation for understanding more complex functions.