91Ó°ÊÓ

Integer functions are special mathematical functions where both the input and output values are whole numbers. In this context, we are looking at a function denoted as \(f: \mathbb{Z} \rightarrow \mathbb{Z}\), which tells us that any integer input \(m\) will yield another integer as the output \(f(m)\).
For instance, let's consider the function \(f(m) = 3 - m\). This equation reveals that for every integer \(m\) you plug into the function, you’ll subtract \(m\) from 3 to find the result. Integer functions, such as this one, are great for modeling real-world scenarios where only whole numbers make sense, like counting objects or steps.
Understanding integer functions helps build a solid foundation in maths, enabling one to handle problems involving sequences, series, and discrete structures efficiently.

The range of a function describes all potential results it can produce. In simpler terms, it's the set of all possible outputs.
For the function \(f(m) = 3 - m\), since \(m\) can be any integer \(\mathbb{Z}\), the resulting outputs will also be integers. Therefore, the range of this function is all integers \(\mathbb{Z}\).
This occurs because subtracting an integer from 3 can result in any other integer. When dealing with function ranges in general, it’s essential to consider all possible ways the input domain affects the outputs.

Function preimages are about finding the set of all inputs that can produce a specific output when fed into the function. It's like working backward from the output to determine which inputs are possible.
For the function \(f(m) = 3 - m\), let’s find the preimage of 5 and 4. To do this, solve the equations for \(m\):

• For the preimage of 5: Set \(f(m) = 5\), leading to \(3 - m = 5\). Solving gives \(m = -2\), so \(-2\) is the preimage of 5.
• For the preimage of 4: Set \(f(m) = 4\), leading to \(3 - m = 4\). Solving gives \(m = -1\), so \(-1\) is the preimage of 4.

By understanding preimages, one gains insights into how inputs and outputs are connected, providing a deeper understanding of a function's behavior.

Graphing discrete functions involves plotting points that correspond to integer inputs rather than drawing a continuous line.
For the function \(f(m) = 3 - m\) with integer inputs, we plot only those points where both input and output are integers. This results in a series of isolated points lined up along a line with a slope of \(-1\).
To sketch this graph:

• Mark the y-intercept, which is the point \((0, 3)\).
• Plot additional points at coordinates \((-7, 10), (-3, 6), (3, 0)\), and \((7, -4)\).
• These points align with a line's path and illustrate the linear progression, yet, they are discreet, not forming a continuous line.

Graphing discrete functions is vital for visualizing them and understanding the relationship between inputs and outputs when only specific values are applicable.