Properties of Functions

Published: Jan 2nd, 2021

Here, we will cover the basic properties of functions that you'll need to know. The properties of functions is an over-arching topic in Grade 12 Functions, so it's a good idea to really these properties early on.

Symmetry

A function can have even, odd, or no symmetry.

  • Even symmetry refers to the idea that a function is symetrical about the y-axis. So when you flip the function about the line \(x=0\), you get the same function. If you remember Lesson 5: Transformations of Functions from Grade 11 Functions, then you might remember that \(f(-x)\) represents a flip along the y-axis. Given this, a simple rule we can use to test for even symmetry for some function, \(f(x)\), is if \(f(x) = f(-x)\). If \(f(x) = f(-x)\), the function has even symmetry. Consider \(f(x) = x^2\). This basic quadratic function is \(\cup\)-shaped (smiley face), split into two equal halves by the y-axis. It's pretty easy to tell visually when a function is symmetrical along the y-axis, but what if we don't have a graph? Well, then we use our algebraic test by comparing the left side (LS), \(f(x)\) to the right side (RS) \(f(-x)\) (of course, you can interchange these functions when you're solving for yourselves):

\(\begin{align} LS & = f(x) \\ & = x^2 \end{align}\)\(\begin{align} RS & = f(-x) \\ & = (-x)^2 \\ & = x^2 \\ & = LS \end{align}\)

Just like that, we've proven algebraically also verified that the basic quadratic function, \(f(x) = x^2\), has even symmetry as it holds true for the equation \(f(x) = f(-x)\).

  • Odd symmetry is when a function has rotational symetry about the origin. So when you rotate the function 180 degrees (half circle) about the point \((0,0)\), you get the same function. To test if an algebraic function, \(f(x)\), has odd symmetry, we need to verify that \(-f(x) = f(-x)\). Let's try the basic linear function, \(f(x) = x\). This basic linear function passes through the origin from quadrant 3 (\(-x\) and \(-y\), bottom left) to quadrant 1 (\(+x\) and \(+y\), top right). This one isn't so easy to tell visually, but the idea is that if we rotate the function \(180^\circ\) about the origin, we end up with the original function. Let's try to prove odd symmetry using the LS = RS (left side = right side) proof method:

\(\begin{align} LS & = f(x) \\ & = x^2 \end{align}\)\(\begin{align} RS & = f(-x) \\ & = (-x)^2 \\ & = (-1)^2*x^2 \\ & = x^2 \\ & = LS \end{align}\)

We have now algebraically also verified that the basic linear function, \(f(x) = x \), has odd symmetry as it holds true for the equation \(-f(x) = f(-x)\).

Before we describe intervals of increase and decrease, let's review the use of brackets for describing intervals. If you're here from our Functions 11 course, then you might remember Lesson 6: The Domain and Range. While we didn't specifically mention the use of square brackets, it's actually a much more concise way to show the domain and range than our round bracket method: \(\{ \}\). The choice of bracket affects whether the endpoint is within the interval of interest. The use of square brackets, \([ \hspace{1mm} ]\), indicate the endpoint are included in the interval. For instance, we can rewrite \(-1 \leq x \leq 1\) as \(x\in[-1,1]\) or just \([-1,1]\)Parentheses/round brackets, \(( \hspace{1mm} )\), indicate that the endpoint is not a part of the interval. For example, \(-3 \lt y \lt 1\) can be rewritten as \((-3,1)\). Round brackets are also used when using infinity as an endpoint to indicate that the interval continues indefinitely (since nothing can equal infinity; we can only approach infinity).  For instance, \((-\infty,\infty)\), describes the entire number line from negative infinity to positive infinity. We could also just write \(x \in \rm I\!R\). The final interval, \((0,1]\), is one that includes all numbers between 0 and 1, but excluding 0 and including 1. Square and round brackets can also be combined when describing an interval if only one of the endpoints are included. We can rewrite \(0 < x \leq 1\), which includes 1 but not 0, as \((0,1]\).

Intervals of Increase/Decrease

Now that we've cleared that up, let's cover the intervals of increase and decrease. An interval of increase is the interval from the domain along which the function is increasing (i.e., as x increases, \(x \rightarrow +\infty\), y increases\(y \rightarrow +\infty\)). On the other hand, the interval of decrease is the interval from the domain along which the function is decreasing (i.e., as x increases, \(x \rightarrow +\infty\), y decreases\(y \rightarrow -\infty\)). Let's take a look at the basic quadrating function to highlight this concept. Remember that \(f(x) = x^2\) is \(\cup\)-shaped (smiley face). Therefore, the function will decrease from \((-\infty,0)\) and increase from \((0,+\infty)\). But here's a question, what is happening to the function at \(x = 0\)? Is it increasing or decreasing? Well, The function is actually neither increasing nor decreasing. This is a concept we will soon cover and one that will be expanded upon in calculus, but this small extremity of the the parabola, called an inflection point which is at its vertex!

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