A linear function is defined as an equation in the form of f(x) = mx + b. It is read a function of x. If the variable in the equation is y, then the function is f(y). If the variable in the equation is z, then the function is f(z), etc. Some examples of linear functions are as follows:
f(x) = (1/2)x – 4
g(x) = 2x – 13
f(y) = 0.9y + 2.75
Notice that a function doesn’t have to be defined as f(x). From any function, we can find an output value for any input value.
Example: Given the function f(x) = 4x + 12, find the output for an input of 5.
That means we need to find the value of the function at x = 5, also noted as f(5). Substitute 5 in for x to get 4(5) + 12 = 20 + 12 = 32. Therefore f(5), “the function f evaluated at x = 5”, is 32.
Example: Given the function g(x) = -9x + 12, find g(3) and g(-6).
Substitute 3 in for x to get -9(3) + 12 = -27 + 12 = -15. Therefore, g(3) = -15. Likewise, to find g(-6) substitute -6 in for x to get -9(-6) + 12 = 54 + 12 = 66. Therefore, g(-6) = 66.
A nonlinear function is a function which is not represented graphically by a straight line. Some examples of nonlinear functions are f(x) = x2, f(x) = x3 and f(x) = |x|.
Example: Graph the function f(x) = x2.
The function f(x) = x2 is known as the squaring function. To graph the function, we choose values for x and substitute for x in f(x). Notice the values for x and f(x) in the chart below. The values for f(x) ≥ 0 for any value of x since squaring all real numbers gives a positive result. You will notice that the domain is all real numbers and range is all real numbers greater than or equal to zero. The graph of the squaring function is called a parabola. Notice how the graph is identical on both sides of the y-axis. You can see this by substituting values for x and finding f(x) at those numbers:
f(0) = 0
f(1) = 1, f(-1) = 1
f(2) = 4 f(-2) = 4
f(3) = 9, f(-3) = 9
Plot the coordinates on a rectangular coordinate system.
Example: Graph the function f(x) = |x|.
The function f(x) = |x| is known as the absolute value function. To graph the function we choose values for x and substitute for x in f(x). Notice in the table below that f(x) ≥ 0. The domain is all real numbers and the range is all real numbers greater than or equal to zero.
x = -2, f(x) = 2, (x, f(x)) = (-2, 2)
x = -1, f(x) = 1, (x, f(x)) = (-1, 1)
x = 0, f(x) = 0, (x, f(x)) = (0, 0)
x = 1, f(x) = 1, (x, f(x)) = (1, 1)
x = 2, f(x) = 2, (x, f(x)) = (2, 2)
Plot the coordinates on a rectangular coordinate system.
Example: Graph the function f(x) = x3.
The function f(x) = x3 is known as the cubing function. To graph the function we choose values for x and substitute for x in f(x). The domain and range are both all real numbers.
x = -2, f(x) = -8, (x, f(x)) = (-2, -8)
x = -1, f(x) = -1, (x, f(x)) = (-1, -1)
x = 0, f(x) = 0, (x, f(x)) = (0, 0)
x = 1, f(x) = 1, (x, f(x)) = (1, 1)
x = 2, f(x) = 8, (x, f(x)) = (2, 8)
Plot the coordinates on a rectangular coordinate system.
Functions can be shifted up, down, left and right. These shifts are called translations.
If f(x) is a function, then f(x) + k has the same graph as f(x) except it is translated up k units. The function f(x) – k has the same graph as f(x) except it is translated down k units. If f(x) is a function, then f(x + h) has the same graph as f(x) except it is translated h units to the left. The function f(x – h) has the same graph as f(x) except it is translated h units to the right.
Example: Graph g(x) = x3 + 2.
The graph of g(x) = x3 + 2 is the same in shape as the graph of f(x) = x3 except that it is translated 2 units up. If g(x) = x3 – 2, then it has the same graph of f(x) = x3 except that it is translated 2 units down. An alternate method to graph g(x) is to pick values for x and substitute into x in g(x) and plot the points.
Example: Graph g(x) = |x + 2| – 1.
The graph of g(x) = |x + 2| – 1 is the same shape as the graph of f(x) = |x| except that is has multiple translations. The +2 inside the absolute value shows there is a translation of 2 units to the left. The -1 outside of the absolute value shows there is a translation of 1 unit down. An alternate method to graph g(x) is to pick values for x and substitute into x in g(x) and plot the points.
The is the guide I use with students to ease confusion on the topic of linear and nonlinear functions.
