Understanding functions is vital for anyone intending to master calculus or learning mathematical physics. For those who have never heard of the concept of the function, here’s a quick introduction. A function f(x) is a mathematical expression, you can think of it as an input-output system, establishing a connection between one independent variable x and a dependent variable y. We “throw” in a certain value of x, do what the mathematical expression f(x) demands us to do, and get a corresponding value of y in return. Here’s an example of this:
The expression on the right tells us that when given a certain value of x, we need to multiply the square of x by 2, subtract 3x from the result and in a final step add one. For example, using x = 2, the function returns the value:
So this particular function links the input x = 2 to the output y = 3. This is called the value of the function f(x) at x = 2. Of course, we are free to choose any other value for x and see what the function does with it. Inserting x = 1, we get:
So given the input x = 1, the function produces the output y = 0. Whenever this happens, a value of zero is returned, we call the respective value of x a root of the function. So the above function has one root at x = 1. Let’s check a few more values:
The first line tells us that for x = 0, the value of the function lies on y = 1. For x = -1 we get y = 6 and for x = -2 the result y = 15. What to do with this? For one, we can interpret these values geometrically. We can consider any pair of x and y as a point P(x / y) in the Cartesian coordinate system. Since we could check every x we desire and calculate the corresponding value of y using the function, the function defines a graph in the coordinate system. The graph of the above function f(x) will go through the points P(2 / 3), P(1 / 0), P(0 / 1), P(-1, 6) and P(-2 / 15). Here’s the plot:
Graph of f(x) = 2x² – 3x + 1
You can confirm that the points indeed lie on the graph by following the x-axis, as usual the horizontal axis, and determining what distance the curve has from the x-axis at a certain value of x. For example, to find the point P(2 / 3), we move, starting from the origin, two units to the right along the x-axis and then three units upwards, in direction of the y-axis. Here we meet the curve, confirming that the graph includes the point P(2 / 3). Make sure to check this for all other points we calculated. Of course, to produce such a neat plot, we need to insert a lot more than just six values for x. This uncreative work is best done by a computer. Feel free to check out the easy-to-use website graphsketch.com for this, it doesn’t cost a thing and requires no registration. Note that the plot also shows a second root at x = 0.5, the point P(0.5 / 0). Let’s make sure that this value of x is a root of our function f(x) by inserting x = 0.5 and hoping that it produces the output y = 0:
As expected. This was all very mathematical, but what practical uses are there for functions? We can use them to establish a connection between the value of one physical quantity and another. For example, through experiments or theoretical considerations we can determine a function f(p) that links the air pressure p to the air density D. It would allow us to insert any value for the air pressure p and calculate the corresponding value for the air density D, which can be quite useful. Or consider a function f(v) that establishes a connection between the velocity v of an object and the frictional forces F it experiences. This is extremely helpful when trying to determine the trajectory of the object, yet another function f(t) that specifies the link between the elapsed time t and the position x of the object. Just to give you one example of this, the function:
connects the elapsed time t (in seconds) with the corresponding height h (in meters) for an object that is dropped from a 22 m tall tower. According to this function, the object will have reached the following height after t = 1 s:
Insert any value for t and the function produces the object’s location at that time. In this case we are particularly interested in the root. For which value of the independent variable t does the function return the value zero? In other words: after what time does the object reach the ground? We could try inserting several values for t and hope that we find the right one. A more promising approach is setting up the equation f(x) = 0 and solving for x. This requires some knowledge in algebra.
Subtracting 22 on both sides leads to:
Divide by -4.91:
And apply the square root:
For this value of t the function f(t) becomes zero (due to inevitable rounding errors, not perfectly though). The rounding errors are also why I switched from the “is equal to”-sign = to the “is approximately equal to”-sign ≈. You should do the same in calculations whenever rounding a value.
Graph of f(t) = -4.91t² + 22
As you can see, functions are indeed quite useful. If you had trouble understanding the algebra that led to the root t = 2.12, consider reading my free e-book “Algebra – The Very Basics” before continuing with functions.
This was an excerpt from my e-book Math Shorts – Exponential and Trigonometric Functions