The function concept
The concept of a function is one of the most important in all of mathematics. A function is an abstraction - an idea or a way of seeing a concept rather than the thing itself. That may seem strange, but it's really pretty easy and useful.
Functions are a way of putting many different concepts on a more even playing field. Using functions, you'll see that many kinds of mathematical constructs have a few identical features, and that makes them all easier to understand. Always remember:
More things are similar in mathematics than different.
Much of the time in mathematics we'll think about functions rather than equations. A function can be thought of as a "machine" or a "black box" into which we put an independent variable, say x, and out of which we get a dependent variable, say y. The function itself is just a set of instructions for what do do with x in order to get the one (and only one) value of y that goes with it.
The two form an ordered pair (x, y), the coordinates of a point on the Cartesian plane that is on the graph of the function.
For example, we could have a function that takes x as input, squares it, and gives back the result (the output) as y. We might write this function as y = x2.
So if we put a 2 into our squaring "machine", it would give us back a 4. If we put in a 3, it would give us back a 9, and so on. Each x we put in is paired with a y that results from following the instructions for what to do with x.
Cartesian coordinates
Cartesian coordinates are the normal 2-dimensional (2D) or 3-dimensional (3D) coordinate systems we most-frequently use. In two dimensions, we draw x- and y-axes at 90˚ angles to each other, and in 3D we add a third axis, usually the z-axis, perpendicular to the x-y plane.
The location or direction of an point or particle can be described using Cartesian coordinates (x, y) in the 2D plane, or (x, y, z) in 3D.
Dependent & Independent variables
The value of the dependent variable depends on the value of the independent variable.
The independent variable is the input to a function (often x); the dependent variable is the output (often y).
A function, given a value of the independent variable, x, generates a point (x, y) in the Cartesian plane
Function notation
We'll need a way to write and refer to functions. For now, let's stick with our squaring function. The usual way to write that function is shown here:
It's important to remember that this is just a special kind of notation for naming functions. It does not mean a variable f multiplied by the variable x. “f of x” is a function that takes in the variable x and squares it to give a corresponding y. In this context, f is not a variable.
Very often, to distinguish different functions we might be working with at the same time, we might write them as f(x), g(x), h(x), and so on, meaning “f of x”, “g of x” and “h of x.”
Here are two more examples:
$$ \begin{align} g(x) &= x^3 - 4 \\[5pt] h(x, y) &= x^2 - 2^y \end{align}$$
Here, the function g takes a single independent variable, x. It calculates the cube of x, subtracts 4 from it and returns the result to you, the user. The function h takes as its input two independent variabless, x and y. It squares x and subtracts 2 raised to the power of y, and returns the result.
The "mapping" concept
Sometimes we say that a function provides us with a mapping of one set of numbers, the domain (the x's) onto another set of numbers, the range (the y's). The domain and range of a function are sets of numbers, and they can be different.
The function tells us which y in the range goes with each x in the domain. For a function to be valid, it must be single-valued. That is, each x in its domain gets paired with one, and only one y. We will learn more about the domain below.
"Parent" functions
It is useful to think of many different classes of functions in their simplest form, and then learn how to transform those functions into more complicated forms. You should become familiar with the forms of these ten "parent functions" below. Don't worry too much if you're unfamiliar with some of them. You'll get to them all eventually.
Click on the graph of each to go directly to the notes for that kind of function. The information to the right of each function is just a little bit to get you familiar; there's much more to know about each one.
You don't have to be familiar with all of the kinds of functions in the panels below. Many you have probably not even encountered. You will eventually, so you might want to check back here once in a while to see how they fit together.
The graph of a linear function (surprise!) is a line. Lines are completely determined by knowing where one point is, and by knowing the tilt or slope.
Functional form: f(x) = mx + b
or y = mx + b, which is called slope-intercept form. It can be rewritten using a little algebra to the standard form Ax + By = C, where A, B and C are constants. This is really just what we get when we move all variables to the left side, all constants to the right.
The graph of a quadratic function (and only a quadratic function) is a parabola.
The parabola is also one of the conic sections.
Parameters: vertex location, direction of opening and steepness of rise or drop.
Function form: f(x) = Ax2 + Bx + C, where
A, B and C are constants.
The graph of a cubic function is (sideways) S-shaped or "sigmoid." The parent function has inversion symmetry about (x, y) = (0, 0). That point is called an inflection point, the point where the curvature of the graph changes sign.
Functional form: f(x) = Ax3 + Bx2 + Cx + D,
where A, B, C and D are constants. Cubic and higher functions are generally referred to as polynomial functions, though linear and quadratic functions are polynomials, too.
A rational function is a ratio of polynomial functions. Because these functions have variables in the denominator, the denominator can approach zero for certain values of x, which leads to asymptotic behavior. In this example, both of the graph axes are asymptotes
Graphs of rational functions are characterized by asymptotes, and sometimes holes in the graph.
Root functions have an exponent of the independent variable that is less than one.
Root functions are characterized by rapid growth for small values of x, with slower growth as x increases.
Note that the domain (the "allowed" values of x) of a root function with an even exponent (like 1/2) is always [0, ∞), while an odd exponent (like 1/3) gives (-∞, ∞).
In an exponential function, the independent variable is in the exponent, thus exponential functions grow very rapidly compared to polynomial functions.
Exponential functions are used to model things that grow, like populations or investments. They are one of the more useful kinds of mathematical functions.
Notice that the rate of growth of an exponential function increases as x grows.
Logarithmic functions are the inverses of exponential functions. They grow continuously as the independent variable grows, but the rate of growth diminishes as x grows.
"Log" functions will be extremely useful in solving some kinds of problems involving growth, like interest in the financial world or population growth. Stay tuned.
Logistic functions are a special class of exponential function. They are constructed so that they model the fact that no real population can grow exponentially indefinitely. All real populations encounter some growth pressure such as competition for food and habitat. Logistic functions grow exponentially, but at some point that growth approaches (asymptotically) a limit.
Logistic functions are also rational functions.
Transformations of functions
The concept of transforming a parent function to alter its shape is a key theme in mathematics and in science, where we use functions to model real things. By transformation, we mean changing the function to alter the shape of its graph arbitrarily (any way we want). For example, through the appropriate set of transformations, the parent-function parabola, y = x2 can be transformed into any other parabola.
There are four important kinds of transformations that you need to know:
- Vertical translation
- Horizontal translation
- Vertical scaling
- Horizontal scaling
To translate something (mathematically) is to move it up or down, left or right. To scale means to vertically or horizontally stretch or compress. As you look through the figures below, note that stretching in one direction is the same as compression in the other, and vice-versa. Let's look at each of these and see how it affects the graph of a function.
Transformations
To translate something means to move it back and forth or up and down, to move it in one dimension.
To scale something means to stretch or compress it along a direction.
Vertical translation
Consider the parent function of the parabolas, f(x)=x2, the black curve on the right. It's not too difficult to imagine that if we simply add a constant number (it's 2 in the figure) to every value of the function, we just raise (translate) the curve upward along the y-axis by 2 units.
The opposite is true when we subtract 2 from the function: It moves downward by 2 units.
That's our first transformation: If f(x) becomes f(x) + k, that means we've just translated the whole graph upward by k units, where a positive k means up and a negative k means down. (It's common to use the letter k for this; I don't know why).
$$f(x) \; \rightarrow \; f(x) + \bf k$$
k > 0 raises the function graph by k units.
k < 0 lowers the function graph by k units.
Horizontal translation
When a number, usually denoted by h, is subtracted from the independent variable inside of a function, the function is translated by h units to the right if h > 0 and to the left if h < 0.
This can be tricky. Remember that the transformation is written f(x)→f(x - h); the h is subtracted. When h is positive, the translation is in the positive x direction. When h is negative, (x-(-h)) = (x+h), and the translation is to the left. For example,
$f(x) = (x - 2)^2$ is a parabola translated to the right by two units.
$f(x) = (x + 2)^2$ is a parabola translated to the left by two units.
$$f(x) \; \rightarrow \; f(x - \bf{h})$$
h > 0 translates the function graph to the right by h units.
h < 0 translates the function graph to the left by h units.
Combined translations
Any set of translations can be combined in a single function. Here, the two translations shown above have been applied to the same function.
Combined translations: $f(x) = (x - h)^2 + k$
They say to take the parent function f(x), and move it h units to the right, and k units upward. These transformations work in any function at all, as you will see as you build up your own function library.
Some examples of translated functions
| $f(x) = x^2$ $g(x) = (x - 4)^2 - 2$ |
The quadratic function f(x) = x2 moved right by 4 units and downward by 2 to form the new function g(x) |
| $f(x) = x^3$ $g(x) = (x + 1)^3 + 5$ |
The cubic function moved left by 1 unit and upward by 5 |
| $f(x) = 2^x$ $g(x) = 2^{(x - 3)} - 7$ |
The exponential function y = 2x, shifted to the right by 3 units and translated downward by 7 units |
Vertical scaling
When a function is multiplied by a constant, usually denoted by A, the result is vertical scaling of the graph. In this case, f(x) becomes A·f(x)
When A > 1, the graph is stretched vertically. Compare the orange graph, $f(x) = 2x^2,$ to the parent function in black.
When 0 < A < 1, the graph is compressed vertically (purple compared to black), and
when A < 0, the graph is flipped or reflected across the x-axis (shown further down the page).
$$f(x) \; \rightarrow \; \bf{A} f(x)$$
A > 1 stretches the function graph vertically.
0 < A < 1 shrinks the function graph vertically.
In either case, if A is negative, the function graph is reflected across the x-axis.
Horizontal scaling
When the independent variable is divided by a constant, usually denoted by c, the result is scaling of the graph along the x-axis.
When c > 1, the graph is stretched horizontally (made wider).
When 0 < c < 1, the graph is compressed (made smaller) horizontally, and
when c < 0, the graph is reflected across the y-axis (and stretched or compressed depending on the absolute value of c).
Notice that horizontal scaling can accomplish the same thing as vertical scaling: H-compression = V-stretching. However, because the parameter c is acted upon by the function (squared in the cases in the graph), the result is slightly different: The graphs of $f(x) = 4x^2$ and $g(x) = (x/0.5)^2$ are the same.
$$f(x) \; \rightarrow \; f\left(\frac{x}{ \bf{c} }\right)$$
c > 1 stretches the function graph horizontally.
0 < c < 1 shrinks the function graph horizontally.
In either case, if c is negative, the function graph is reflected across the y-axis.
Transformations: putting it all together
The figure above shows a generic function f(x) with all four transformations present. These transformations can take a basic function and stretch and bend it into a form that could model something real.
For example, we might want to model the hanging cables on the Golden Gate Bridge using an upside-down parabola. To do that we could take the parent function, flip it upside-down and stretch it using the A parameter, stretch it horizontally using the c parameter, and move it into place, perhaps setting up the origin at the center of the bridge, using the translation parameters, h and k.
The most important thing about these transformations is that they do the same things no mater what the function, whether it's a linear function or a trigonometric one (you'll learn about those later if you don't know them now.
The beauty of these transformations is that they work the same on any function.
The absolute-value function is a nice way to learn transformations.
The simple shape of the absolute-value function (the black V in the graphs below) makes it a good model function for studying transformations of all kinds. Here are a few:
Translation
In the top plot, the constant 2 has been added and subtracted to the independent variable before the function is evaluated. $f(x) = |x - 2|$ is the parent function shifted two units to the right (orange), and $f(x) = |x + 2|$ is a shift of two units to the left (purple).
In the bottom graph, the parent function of absolute value functions, $f(x) = |x|$ (black) is compared to
$$f(x) = |x| - 2 \; \text{ and } \; f(x) = |x| + 2$$
in orange and purple, respectively. Notice that adding or subtracting a constant to a function just raises or lowers the value it returns by that amount. This is vertical translation.
The horizontal translation seems backward, so be careful y = |x - h| is a shift of h units to the right when h is positive. When h is negative, the function would be written as y = |x + h| and the shift would be to the left.
Scaling & reflection
In the top figure, we multiply the absolute value parent function f(x) = |x| by 2 to stretch it vertically (orange), and by ½ to compress it vertically (purple). The function g(x) = |x/0.5| would be equivalent to the first – a two-fold horizontal compression being equivalent to a two-fold vertical stretch. Likewise, g(x) = |x/2| would yield the purple curve.
When the scaling factor is negative (lower graph), the function is reflected across the x-axis, but also scaled by the absolute value of the scaling factor.
The orange plot shows f(x) = |x| reflected across the x-axis, and the purple plot is the same function reflected across the x-axis and scaled (stretched) vertically by a factor of 2.
You've just begun to know all there is to know about functions. Now you need to figure out how to determine the domain and range of a function, how to put functions together in compositions that will allow you to solve a huge number of problems, and how to find inverses of functions – functions that undo the action of other functions. Use the links below or the menu at the top of this page to navigate onward. Have fun!
Practice problems
Sketch a graph of each of the absolute value functions. Approach the problems by doing one transformation at a time, starting with the parent function, f(x) = |x|.
| 1. | $f(x) = |x - 7|$ | |
| 2. | $f(x) = |x + 3| - 2$ | |
| 3. | $f(x) = -|x - 2| + 2$ | |
| 4. | $f(x) = \frac{1}{2}|x + 1| - \frac{1}{2}$ |
| 5. | $f(x) = -3|x| + 4$ | |
| 5. | $f(x) = \big|\frac{x}{2} + 3 \big| - 1$ | |
| 7. | $f(x) = -|3x| - 4$ | |
| 8. | $f(x) = -\frac{1}{2} |2x + 3| + 3$ |
Problem 1 solution
Problem 2 solution
Problem 3 solution
Problem 4 solution
Problem 5 solution
Problem 6 solution
Problem 7 solution
Problem 8 solution
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