1.03 Domain, range, zeros, and symmetry

Domain and range

Recall that the domain of a relation is the set of all possible input values for a relation, while the range denotes the set of all possible output values.

Graphically

We can use the graph of a function to determine its domain and range. Let's consider the graph below:

Horizontally this graph spans from $-1$−1 to $1$1, so we can write the domain as $-1\le x\le1$−1≤x≤1. Similarly, the graph goes vertically from $-2$−2 to $2$2 so the range can be written as $-2\le y\le2$−2≤y≤2. 

Think about a straight line like $y=2x+1$y=2x+1, pictured below. Its domain and range have no restrictions at all. And so, we could write the domain and range in interval notation as $\left(-\infty,\infty\right)$(−∞,∞).

 

 

Practice questions

Question 1

Question 2

 

Intercepts

A point where a graph intersects an axis is called an intercept. The $y$y-intercept of a function occurs at the point where the $x$x-value of the function is zero. The $x$x-intercept occurs where the $y$y-value is zero. By definition, the graph of a function can have $0$0, $1$1, or many $x$x-intercepts. However, it can only have one $y$y-intercept.

Practice questions

Question 3

Question 4

 

Zeros

We can write down the equation of a function in the form $y=f\left(x\right)$y=f(x). For example, let's consider the quadratic function with equation $y=x^2-1$y=x2−1. This equation has infinitely many solutions, such as the pair $x=2,y=3$x=2,y=3, with each $x$x-value corresponding to a $y$y-value.

Now, there is a particular set of solutions which have $y=0$y=0. We call the corresponding values of $x$x the zeros of the function. For some functions, such as $y=x^2-1$y=x2−1, we can set the $y$y-value to be $0$0 and solve the resulting equation to find the zeros. For other functions, such as $y=e^x-3x^2$y=ex−3x2, it is not so easy to find the zeros algebraically.

 

Finding the number of zeros graphically

In many situations, it is useful to know how many zeros a function has, without necessarily finding the particular values. To do so, we can make use of the graph of the function.

Here is the graph of a function $y=f\left(x\right)$y=f(x):

The zeros of a function are the values of $x$x which correspond to $y=0$y=0. Looking at the graph, we can see that all points which have a $y$y-coordinate of $0$0 lie on the $x$x-axis. That is, the zeros of a function are the $x$x-coordinates of the $x$x-intercepts of the curve!

For the function in the graph above we can see that the curve crosses the $x$x-axis three times, so this function has three zeros. The intercepts are highlighted in the following graph:

Note that not all functions have zeros! This can also be seen by looking at a graph of the function. If the curve does not intersect the $x$x-axis at any point, then there are no values of $x$x which correspond to $y=0$y=0 and so the function has no zeros. Here is the graph of $y=x^2+1$y=x2+1 as an example:

 

 

Finding the number of solutions to an equation

We can also use this technique when looking for the number of solutions to an equation, such as $x^3=x-3$x3=x−3.

To do so, we first rearrange the equation so that one side is equal to zero:

$x^3-x+3=0$x3−x+3=0

The solutions to this equation are the zeros of the function $y=x^3-x+3$y=x3−x+3. So we can determine the number of solutions by sketching a graph of $y=x^3-x+3$y=x3−x+3 (using technology or otherwise) and looking at the number of $x$x-intercepts:

Looking at the graph we can see that this curve has one $x$x-intercept, and so the original equation $x^3=x-3$x3=x−3 has exactly one real solution.

Practice questions

Question 5

Question 6

 

Symmetry

Functions can be grouped into three varieties. There are odd functions, there are even functions and there are functions which are neither odd nor even. The reason for the classification of oddness and evenness of a function has to do with symmetry. Knowing that a function is odd or even assists us with understanding more about the shape of a function's graph.  

 

Even functions

An even function $f\left(x\right)$f(x) exhibits the property that, for all $x$x values in the domain,  $f\left(-x\right)=f\left(x\right)$f(−x)=f(x). 

For example the function $f\left(x\right)=x^4-8x^2+16$f(x)=x4−8x2+16 is even because we can show that:

$f\left(-x\right)$f(−x)	$=$=	$\left(-x\right)^4-8\left(-x\right)^2+16$(−x)4−8(−x)2+16
	$=$=	$x^4-8x^2+16$x4−8x2+16
	$=$=	$f\left(x\right)$f(x)

An even function exhibits reflective symmetry across the $y$y-axis as shown in the graph below. 

 

Odd functions 

Algebraically speaking, an odd function $f\left(x\right)$f(x) exhibits the property that, for all $x$x values in the domain, $f\left(-x\right)=-f\left(x\right)$f(−x)=−f(x). 

So for example the function given by $f\left(x\right)=x^3-x$f(x)=x3−x is odd because:

$f\left(-x\right)$f(−x)	$=$=	$\left(-x\right)^3-\left(-x\right)$(−x)3−(−x)
	$=$=	$-x^3+x$−x3+x
	$=$=	$-\left(x^3-x\right)$−(x3−x)
	$=$=	$-f\left(x\right)$−f(x)

It is very important that your setting out is as perfect as the worked example shown above if you are asked to algebraically show that a function is odd. You might have been able to spot that the function was odd when you had completed the second line of working, with $f(-x)=-x^3+x$f(−x)=−x3+x. However, the step to show the factoring that follows is crucial to convince someone marking your solution that you are certain of the property that $f(-x)=-f(x)$f(−x)=−f(x). 

 

As a graph, an odd function possesses rotational symmetry. Specifically, this means that an odd function can be rotated $180^\circ$180°  about the origin and fall back onto itself. The graph of $f\left(x\right)=x^3-x$f(x)=x3−x is shown below. Note, from the example depicted in the diagram that any specific value of $x$x exhibits the oddness property that $f\left(-x\right)=-f\left(x\right)$f(−x)=−f(x). 

 

 

If a function isn't odd or even, then it is said to be neither, and many functions we encounter are in this last category. This is particularly true for functions that seem to have symmetry but not in the exact manner we are seeking. An example is the graph shown below. Its rotational symmetry isn't centered on the origin. Looking closely at the $y$y values of the curve, it is clear that $f(-x)\ne-f(x)$f(−x)≠−f(x). In fact, you might have realized the important condition that if a continuous function is going to be odd, it must pass through the origin.

 

 

Examples of odd and even functions

We can suspect oddness when the powers of $x$x in the function are all odd. For example, $f\left(x\right)=x^3$f(x)=x3, and $f\left(x\right)=x^5-2x^3+7x$f(x)=x5−2x3+7x are all odd. We suspect evenness when the powers of $x$x in the function are all even. For example, $f\left(x\right)=x^6-2$f(x)=x6−2, $f\left(x\right)=x^2$f(x)=x2, and $f\left(x\right)=3x^8-5x^4$f(x)=3x8−5x4 are all even. 

But there are other functions to consider beside polynomials. For example, the hyperbola $y=\frac{k}{x}$y=kx​ (for some constant $k$k) is odd because, for all $x$x in the domain,  $\frac{k}{\left(-x\right)}=-\frac{k}{x}$k(−x)​=−kx​. The semicircle $y=\sqrt{r^2-x^2}$y=√r2−x2 for some radius $r$r is even because $\sqrt{r^2-\left(-x\right)^2}=\sqrt{r^2-x^2}$√r2−(−x)2=√r2−x2. 

 

 

Practice Questions

QUESTION 7

QUESTION 8

QUESTION 9