Now that we’ve discussed the functions which correspond to horizontal lines, \(y = b\), we move to discussing the functions which can be represented by lines of the form \(y = mx + b\) where \(m \neq 0\). These functions are called linear functions and are described below.

As with Definition constantfunction, in Definition linearfunction, \(x\) is the independent variable, \(f\) is the function name, and both \(m\) and \(b\) are parameters. Notice that \(m\) is restricted by \(m \neq 0\) for if \(m = 0\) then the function \(f(x) = mx + b\) would reduce to the constant function \(f(x) = b\). The domain of linear functions, like that of constant functions, is specified as \((-\infty , \infty )\)

Recall that the form of the line \(y = mx + b\) is called the slope-intercept form of the line and the slope, \(m\), and the \(y\)-intercept \((0, b)\), are easily determined when the line is written this way. Likewise, the form of the function in Definition linearfunction, \(f(x) = mx + b\), is often called the slope-intercept form of a linear function.

The graph of a linear function is the graph of the line \(y = mx + b\). Lines are uniquely determined by two points, and two points of geometric interest are the axis intercepts. We’ve already reminded you of the \(y\)-intercept, \((0,b)\), which is obtained by setting \(x = 0\). Similarly, to find the \(x\)-intercept, we set \(y = 0\) and solve \(mx + b = 0\) for \(x\). We leave this to the reader in Exercise xinterceptoflinear. In addition to having special graphical significance, axis intercepts quite often play important roles in applications involving both linear and non-linear functions. For that reason, we take the time to define them here using function notation.

As is customary in this text, Definition interceptdefns uses the default independent variable \(x\), function name \(f\), and dependent variable \(y\), so these letters will change depending on the context. Also note that the ‘zeros’ of a function are the solutions to \(f(x) = 0\) - so they are real numbers. The \(x\)-intercepts are, on the other hand, points on the graph. As a quick example, consider \(f(x) = x-3\). The zeros of \(f\) are found by solving \(f(x) = 0\), or \(x-3=0\). We get one solution, \(x = 3\). Therefore, \(x=3\) is the zero of \(f\) that corresponds graphically to the \(x\)-intercept \((3,0)\).

We now turn our attention to slope. The role of slope, or more generally a ‘rate of change’, in Science and Mathematics cannot be overstated. As you may recall, or quickly read about on page ??, the slope of a line that has been graphed in the \(xy\)-plane is defined geometrically as follows:

\[m = \frac {\text {rise}}{\text {run}} = \frac {\Delta y}{\Delta x} ,\]

where the capital Greek letter ‘\(\Delta \)’ denotes ‘change in.’ In this course, it is vital that we regard the slope of a linear function as a rate of change of function outputs to function inputs. That is, given the graph of a linear function \(y = f(x) = mx + b\):

\[ m = \frac {\text {rise}}{\text {run}} = \frac {\Delta y}{\Delta x} = \frac {\Delta [f(x)]}{\Delta x} = \frac {\Delta \text {outputs}}{\Delta \text {inputs}}. \]

What is important to note here is that for linear functions, the rate of change \(m\) is constant for all values in the domain. We’ll see the importance of this statement in the upcoming examples.

Geometrically, the sign of the slope has a profound impact on the graph of the line. Recall that if the slope \(m > 0\), the line rises as we read from left to right; if \(m<0\), the line falls as we read from left to right; if \(m=0\), we have a horizontal line and the graph plateaus. We define these notions more precisely for general functions in the following definition.

Again, as with Definition interceptdefns, Definition incdeccnstdefn applies to any function, not just linear and constant functions. Also, note that, like Definition absmaxmindefn, Definition incdeccnstdefn blurs the line between the function, \(f\), and its outputs, \(f(x)\), because the verbiage ‘\(f\) is increasing’ is really a statement about the outputs, \(f(x)\). Finally, when we ask ‘where’ a function is increasing, decreasing or constant, we are looking for an interval of inputs. We’ll have more to say about this in later sections, but for now, we summarize these ideas graphically below.

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From the graphs above, we see that regardless if \(m>0\) or \(m<0\), the range of linear functions is \((-\infty , \infty )\). Therefore, linear functions have no maximum or minimum.

A couple of remarks about Example PortaBoyCost are in order. First, if \(x\) represents the number of PortaBoy game systems being produced, then \(x\) can really only take on whole number values. We will revisit this scenario in Section QuadraticFunctions where we will see how the approach presented here allows us to use more elegant techniques when analyzing the situation than a discrete data set would allow.

Second, once we know that the variable cost is \(\$80\) per PortaBoy, we can revisit a computation we did earlier in the example. We computed \(C(185) = 14950\) and needed to compute \(C(186)\). With \(186\) being just one more PortaBoy than \(185\), we can use the variable cost to get

\[C(186) = C(185) + 80(1) = 14950 + 80 = 15030,\]

which agrees with our earlier computation. If we wanted to find \(C(300)\), we could do something similar. Using \(300 - 185 = 115\), we can find \(C(300)\) as follows:

\[C(300) = C(185) + 80(115) = 14950 + 9200 = 24150.\]

In general, we could rewrite \(C(x) = C(185) + 80(x - 115)\). This same reasoning shows that for any \(x_0\) in the domain of \(C\), we have \(C(x) = C(x_0) + 80(x - x_0)\) - a fact we invite the reader to verify.

Indeed, the computations above are at the heart of what it means to be a linear function: linear functions change at a constant rate known as the slope. To better see this algebraically, recall that given a point \((x_0, y_0)\) on a line along with the slope, \(m\), the point-slope form of the line is: \(y - y_0 = m(x - x_0)\). Rewriting, we get \(y = y_0 + m (x - x_0)\) and setting \(y = f(x)\) and \(y_0 = f(x_0)\) yields:

A few remarks are in order. First note that if the point \((x_0, f(x_0))\) is the \(y\)-intercept \((0, b)\), Equation linearfunctionpointslope immediately reduces to the slope-intercept form of the line: \( f(x) = f(x_0) + m (x - x_0) = b + m(x - 0) = mx + b,\) so you can use Equation linearfunctionpointslope exclusively from this point forward.

Second, if we write \(\Delta x = x - x_0\), then \(x = x_0 + \Delta x\) so we can rewrite Equation linearfunctionpointslope as follows:

\[ \begin{array}{ccccc} f(x_0 + \Delta x) & = & f(x_0) & + & m \Delta x \\ (\text {new output}) & = & (\text {known output}) & +& (\text {change in outputs}) \\ \end{array} \]

In other words, changing the input by \(\Delta x\) results in changing the output by \(m \Delta x\). This tracks since

\[ m \Delta x = \frac {\Delta [f(x)]}{\Delta x} \Delta x = \Delta [f(x)] = \Delta \text {outputs}. \]

The fact that we can write \(\Delta \text {outputs} = m \Delta x\) for any choice of \(x_0\) is another way to see that for linear functions, the rate of change is constant. That is, the rate of change, \(m\), is the same for all values \(x_0\) in the domain. We’ll put Equation linearfunctionpointslope to good use in the next example.

The function \(p\) in Example PortaBoyDemand is called the price-demand function (or, sometimes called more simply a ‘demand function’) because it returns the price \(p(x)\) associated with a certain demand \(x\) - that is, how many products will sell. These functions, along with cost functions like the one in Example PortaBoyCost, will be revisited in Example PortaBoyProfit.

Our next two examples focus on writing formulas for piecewise-defined functions, the second of which models a real-world situation.