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The process of graphing linear functions begins with understanding what a linear function is. A linear function is a function whose graph in a coordinate plane is a straight line. These functions can be written in the form of an equation, such as f(x) = mx + b where m represents the slope, and b represents the y-intercept—the point at which the line crosses the y-axis.

To graph a linear function, you should identify at least two points on the line by plugging in different x-values into the function and solving for y. Then, plot these points on a coordinate plane and draw a straight line through them. For instance, with f(x) = x + 1, plugging in x-values of 0 and -1 yields points (0, 1) and (-1, 0), respectively. Drawing a line through these points will produce the graph of the function.

The domain of a function refers to all of the possible input values (usually x-values) for which the function is defined. In simpler terms, it is the set of x-values you can plug into your function to get out a real number. For linear functions, such as f(x)=x+1, there are no restrictions on x-values, as they can take on any real value from negative to positive infinity. Thus, the domain is all real numbers, which is denoted as \( (-\text{\infty}, +\infty) \). When finding the domain, consider any potential restrictions such as fractions (which cannot have a zero denominator) or square roots (which must have a non-negative radicand), but with linear functions, these restrictions do not apply.

The range of a function, in contrast to the domain, signifies all possible outputs or y-values that can result from plugging the domain into the function. In the case of a linear function without any horizontal segments, like our example of f(x) = x + 1, the y-values can include any real number because as you choose increasingly larger or smaller x-values, the y-value will adjust accordingly. Therefore, the range is also all real numbers, written as \( (-\text{\infty}, +\infty) \). Always remember to look for any constraints that might limit the range, such as maximum or minimum y-values for functions that are not continuous everywhere.

The slope is a fundamental aspect of linear functions that describes how steep the line is. Algebraically, it's the 'rise over run' or the change in y over the change in x between two points on the line. The slope is the value of m in the standard form y = mx + b. In our examples, f(x)=x+1 and g(x)=x-1, both slopes are 1. This slope means that for each single unit of increase in x, y increases by the same amount, hence the '1' for 'rise' and 'run'. A slope of 0 indicates a horizontal line, and undefined slope (where division by zero would occur) indicates a vertical line. Graphically, if the line moves upwards as it goes from left to right, the slope is positive; if it moves downwards, the slope is negative.