17.3: Appendix C- A Review of Simple Functions

We review a few basic concepts related to functions.

C.1 What is a function?

A function is just a way of expressing a special relationship between a value we consider as the input (" \(x\) ") value and an associated output (" \(y\) ") value. We write this relationship in the form

\[y=f(x) \nonumber \]

to indicate that \(y\) depends on \(x\). The only constraint on this relationship is that, for every value of \(x\) we can get at most one value of \(y\). This is equivalent to the "vertical line property": the graph of a function can intersect a vertical line at most at one point. The set of all allowable \(x\) values is called the domain of the function, and the set of all resulting values of \(y\) are the range.

Naturally, we do not always use the symbols \(x\) and \(y\) to represent independent and dependent variables. For example, the relationship

\[V=\frac{4}{3} \pi r^{3} \nonumber \]

expresses a functional connection between the radius, \(r\), and the volume, \(V\), of a sphere. We say in such a case that " \(V\) is a function of \(r\) ".

All the sketches shown in Figure C.2 are valid functions. The first is merely a collection of points, \(x\) values and associated \(y\) values. The second a histogram. The third sketch is meant to represent the collection of smooth continuous functions - those of most interest to usnin the study of calculus. On the other hand, the example shown in Figure C.1 is not the graph of a function. We see that a vertical line intersects this curve at more than one point. This is not permitted since a given value of \(x\) may have at most one corresponding value of \(y\).

Figure C.1: The above elliptical curve cannot be the graph of a function. The dashed vertical line intersects the graph at more than one point: this means that a given value of \(x\) corresponds to "too many" values of \(y\). If we restrict ourselves to the top part of the ellipse only (or the bottom part only), then we can create a function which has the corresponding graph.

Figure C.2: All the examples above represent functions.

C.2 Geometric transformations

It is helpful to recognize what happens to the graph of a function when the relationship between the variables is changed slightly. Often this is called applying a transformation. In Figure C.3 we illustrate (a) an original function \(f(x)\), (b) the function \(f(x-a)\) which shifts \(f\) to the right along the positive \(x\)-axis by a distance \(a\), and (c) the function \(f(x)+b\) shifting \(f\) up the \(y\)-axis by height \(b\).

Figure C.3: Shifting the graph of a function horizontally and vertically

Figure C.4 illustrates what happens to a function when shifts, scaling, or reflections occur: a function \(y=f(x)\) is shown with a black solid line. On the same graph are superimposed the reflections of this graph about the \(\mathrm{x}\) axis, \(y=-f(x)\) (dashed black), about the \(y\) axis \(y=f(-x)\) (red), and about the \(y\) and the \(x\)-axis, \(y=-f(-x)\) (red dashed). The latter is equivalent to a rotation of the original graph about the origin.

Figure C.4: A function and its reflections about \(x\) and \(y\)-axes.

C.3 Classifying

While life offers amazing complexity, one way to study living things is to classify them into related groups. A biologist looking at animals might group them according to certain functional properties - being warm blooded, being mammals, having fur or claws, or having some other interesting characteristic. In the same way, mathematicians often classify the objects they study (e.g. functions) into related groups.

An example of way to group functions into very broad classes is also shown in Figure C.5.

Figure C.5: Classifying functions according to their properties.

From left to right, the complexity of behavior in this chart grows: at left, we see constant and linear functions; these are "most convenient" or simplest to describe: one or two parameters suffice (e.g. intercept or slope). Further to the right are smooth and continuous functions, while rightmost some more irregular, discontinuous function represents those that are outside the group of the "well-behaved".

In Section C.4 we study examples along this spectrum. Towards this end, we describe properties they share, properties they inherit form their "cousins," and new characteristics that appear at distinct branches.

C.4 Power functions and symmetry

In this section we list some features of each family of power functions.

Even integer powers. For \(n=2,4,6,8 \ldots\) the shape of the graph of \(y=x^{n}\) is as shown in Figure 1.4(a). Note the following characteristics of these graphs:

1. The graphs of all the even power functions intersect at \(x=0\) and at at \(x=\pm 1\). The value of \(y\) corresponding to both of these is \(y=+1\).
2. All graphs have a lowest point - a minimum value - at \(x=0\).
3. As \(x \rightarrow \pm \infty, y \rightarrow \infty\). We equivalently say that these functions are "unbounded from above."
4. The graphs are all symmetric about the \(y\)-axis. This special type of symmetry is of interest in other types of functions (not just power functions). A function with this property is called an even function.

Mastered Material Check

1. What are the coordinates of the three intersection points of even power functions?

Odd integer powers. For \(n=1,3,5,7, \ldots\) and other odd powers, the graphs have shapes shown in Figure 1.4(b) and the following characteristics:

1. The graphs of the odd power functions intersect at \(x=0\) and at \(x=\pm 1\).
2. None of the odd power functions have a minimum value.
3. As \(x \rightarrow+\infty, y \rightarrow+\infty\). As \(x \rightarrow-\infty, y \rightarrow-\infty\). The functions are "unbounded from above and below."
4. The graphs are all symmetric about the origin. This special type of symmetry is of interest in other types of functions (not just power functions). A function with this type of symmetry is called an odd function.

Mastered Material Check

2. What are the coordinates of the three intersection points of odd power functions?

Further properties of intersections

Consider the even and odd functions graphed in Figure C.6. Notice that a horizontal line intersects the graph only once for the odd power but possibly twice for the even power.

Note: we must allow for the case when the line does not intersect at all, or that it intersects precisely at the minimum point.

These observations holds in the case of general even and odd power functions as well.

Figure C.6: The even power functions intersect a horizontal line in up to two places, while the odd power functions intersect such a line in only one place.

A horizontal line has an equation of the form \(y=C\) where \(C\) is some constant. To find where it intersects the graph of a power function \(y=x^{n}\), we solve an equation of the form

\[x^{n}=C . \nonumber \]

To do so, take \(n^{\text {th }}\) root of both sides:

\[\left(x^{n}\right)^{1 / n}=C^{1 / n} . \nonumber \]

Simplifying, using algebraic operations on powers leads to

\[\left(x^{n}\right)^{1 / n}=x^{n / n}=x^{1}=x=C^{1 / n} . \nonumber \]

However, we must allow that there may be more than one solution to Equation (C.1), as shown for some \(C>0\) in Figure C.6. This demonstrates a distinction between odd and even power functions. If \(n\) is even then the solutions to Equation (C.1) are

\[x=\pm C^{1 / n}, \nonumber \]

whereas if \(n\) is odd, there is a single solution,

\[x=C^{1 / n} . \nonumber \]

In general, we can define even and odd functions.

Definition C.1 (Even and Odd Functions)

A function that is symmetric about the \(y\)-axis is said to be an even function. A function that is symmetric about the origin is said to be an odd function.

Even functions satisfy the relationship

\[f(x)=f(-x) . \nonumber \]

Odd functions satisfy the relationship

\[f(x)=-f(-x) . \nonumber \]

Examples of even functions include \(y=\cos (x), y=-x^{8}, y=|x|\) which are all their own mirror images when reflected about the \(y\)-axis. Examples of odd functions are \(y=\sin (x), y=-x^{3}, y=x\). Each of these functions is its own double-reflection (about \(y\) and then \(x\)-axes).

In a later calculus course, when we compute integrals, taking these symmetries into account can help to simplify (or even avoid) calculations.

Optional: Combining even and odd functions

Not every function is either odd or even. However, if we start with symmetric functions, some manipulations can either preserve or reverse the symmetry.

Example C.1

Show that the product of an even and an odd function is an odd function.

Solution

Let \(f(x)\) be even. Then

\[f(x)=f(-x) . \nonumber \]

Let \(g(x)\) be an odd function. Then \(g(x)=-g(-x)\). We define \(h(x)\) to be the product of these two functions,

\[h(x)=f(x) g(x) . \nonumber \]

Using the properties of \(f\) and \(g\),

\[f(x) g(x)=f(-x)[-g(-x)] \nonumber \]

so, rearranging, we get

\[h(x)=f(x) g(x)=f(-x)[-g(-x)]=-[f(-x) g(-x)] . \nonumber \]

but this is just the same as \(-h(-x)\). We have established that

\[h(x)=-h(-x), \nonumber \]

so the new function is odd.

A function is not always even or odd. Many functions are neither even nor odd. However, it is possible to show that given any function, \(y=f(x)\), we can write it as a sum of an even and an odd function. This is left as a challenge for the reader, with the following hint:

Hint: suppose \(f(x)\) is not an even nor an odd function.

Consider defining the two associated functions:

\[f_{e}(x)=\frac{1}{2}(f(x)+f(-x)), \nonumber \]

and

\[f_{0}(x)=\frac{1}{2}(f(x)-f(-x)) . \nonumber \]

Can you draw a sketch of what these would look like for the function given in Figure C.3(a)? Show that \(f_{e}(x)\) is even and that \(f_{0}(x)\) is odd. Now show that

\[f(x)=f_{e}(x)+f_{0}(x) . \nonumber \]

C.5 Inverse functions and fractional powers

Suppose we are given a function expressed in the form

\[y=f(x) . \nonumber \]

This implies that \(x\) is the independent variable, and \(y\) is obtained from it by evaluating a function, i.e. by using the "rule" or operation specified by that function. This mathematical statement expresses a certain relationship between the two variables, \(x\) and \(y\), in which the roles are distinct: \(x\) is a value we pick, and \(y\) is calculated from it.

However, sometimes we can express a relationship in more than one way: as an example, if the connection between \(x\) and \(y\) is simple squaring, then provided \(x>0\), we might write either

\[y=x^{2} \quad \text { or } \quad x=y^{1 / 2}=\sqrt{y} \nonumber \]

to express the same relationship. In other words

\[y=x^{2} \Leftrightarrow x=\sqrt{y} . \nonumber \]

We have used two distinct functions to describe the relationship from two points of view: one function involves squaring and the other takes a square root. We may also notice that for \(x>0\),

\[\begin{aligned} & f(g(x))=(\sqrt{x})^{2}=x, \\ & g(f(x))=\sqrt{\left(x^{2}\right)}=x, \end{aligned} \nonumber \]

i.e. that these two functions invert each other’s effect.

Functions that satisfy

\[y=f(x) \Leftrightarrow x=g(y) \nonumber \]

are said to be inverse functions. We often use the notation \(f^{-1}(x)\) to denote the function that acts as an inverse function to \(f(x)\).

Graphical property of inverse functions

The graph of an inverse function \(y=f^{-1}(x)\) is geometrically related to the graph of the original function: it is a reflection of \(y=f(x)\) about the \(45^{\circ}\) line, \(y=x\). This relationship is shown in Figure C.7(a) for a pair of functions \(f\) and \(f^{-1}\).

Figure C.7: The point \((a, b)\) is on the graph of \(y=f(x)\). If the roles of \(x\) and \(y\) are interchanged, this point becomes \((b, a)\). Geometrically, this point is the reflection of \((a, b)\) about the line \(y=x\). Thus, the graph of the inverse function \(y=f^{-1}(x)\) is related to the graph of the original function by reflection about the line \(y=x\). In (b), the inverse is not a function, as it does not satisfy the vertical line property. In (a), both \(f\) and its reflection satisfy that property, and thus the inverse, \(f^{-1}\) is a true function.

But why should this be true? The idea is as follows: suppose that \((a, b)\) is any point on the graph of \(y=f(x)\). This means that \(b=f(a)\). That, in turn, implies that \(a=f^{-1}(b)\), which then tells us that \((b, a)\) must be a point on the graph of \(f^{-1}(x)\). But the points \((a, b)\) and \((b, a)\) are related by reflection about the line \(y=x\). This is true for any arbitrary point, and so must be true for all points on the graphs of the two functions.

Restricting the domain

The above argument establishes that, given the graph of a function, its inverse is obtained by reflecting the graph in an imaginary mirror placed along a line \(y=x\).

However, a difficulty could arise. In particular, for the function

\[y=f(x)=x^{2}, \nonumber \]

a reflection of this type would lead to a curve that cannot be a function, as shown in Figure C.8. Note: the sideways parabola would not be a function if we included both its branches, since a given value of \(x\) would have two associated y values. To fix such problems, we simply restrict the domain to \(x > 0\), i.e. to the solid parts of the curves shown in Figure C.8. For this subset of the \(x\)-axis, we have no problem defining the inverse function.

Figure C.8: The graph of \(y=f(x)=x^{2}\) (black) and of its inverse function (red). We cannot define the inverse for all \(x\), because the red parabola does not satisfy the vertical line property. However, if we restrict the original function to positive \(x\) values, this problem is circumvented.

Observe that the problem described above would be encountered for any of the even power functions by virtue of their symmetry about the \(y\)-axis but not by the odd power functions. For example,

\[y=f(x)=x^{3}, \quad y=f^{-1}(x)=x^{1 / 3} \nonumber \]

are inverse functions for all \(x\) values: when we reflect the graph of \(x^{3}\) about the line \(y=x\) we do not encounter the problem of multiple \(y\) values. This follows directly from the horizontal line properties that we saw in Figure C.6. When we reflect the graphs shown in Figure C.6 about the line \(y=x\), the horizontal lines are reflected onto vertical lines. Odd power functions have inverses that intersect a vertical line exactly once, i.e. they satisfy the "vertical line property" discussed earlier.

C.6 Polynomials

Recall that a polynomial is a function of the form

\[y=p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} . \nonumber \]

This form is sometimes referred to as superposition (i.e. simple addition) of the basic power functions with integer powers. The constants \(a_{k}\) are called coefficients. In practice, some of these may be zero. We restrict attention to the case where all these coefficients are real numbers. The highest power \(n\) (whose coefficient is not zero) is called the degree of the polynomial.

We are interested in these functions for several reasons. Primarily, we find that computations involving polynomials are particularly easy, since operations include only the basic addition and multiplication.

Features of polynomials

• Zeros of a polynomial: values of \(x\) such that

\[y=p(x)=0 . \nonumber \]

If \(p(x)\) is quadratic (a polynomial of degree 2) then the quadratic formula gives a simple way of finding roots of this equation/zeros of the polynomial. Generally, for most polynomials of degree higher than 5, there is no analytical recipe for finding zeros. Geometrically, zeros are places where the graph of the function \(y=p(x)\) crosses the \(x\)-axis. This fact is exploited in Chapter 5 to approximate the values of the zeros using Newton’s Method.

• Critical Points: places on the graph where the value of the function is locally larger than those nearby (local maxima) or smaller than those nearby (local minima) are of interest to us. Calculus is one of the main tools for detecting and identifying such places.
• Behaviour for very large \(x\) : all polynomials are unbounded as \(x \rightarrow \infty\) and as \(x \rightarrow-\infty\). For large enough values of \(x\), the power function \(y=f(x)=x^{n}\) with the largest power, \(n\), dominates over other power functions with smaller powers, as seen in Chapter 1 . For

\[p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, \nonumber \]

the first (highest power) term dominates for large \(x\). Thus for large \(x\) (whether positive or negative),

\[p(x) \approx a_{n} x^{n} \text { for large } x . \nonumber \]

• Behaviour for small \(x\) : close to the origin, power functions with smallest powers dominate (see Chapter 1 ). Thus, for \(x \approx 0\) the polynomial is governed by the behavior of the smallest (non-zero coefficient) power, i.e.,

\[p(x) \approx a_{1} x+a_{0} \text { for small } x . \nonumber \]