Functional Analysis

What is a Function?

Before we dive into this topic, let us first recall the definition of a function.

Below is an example of a function.

The Domain and Range

When plotting a function, it is essential to know the 'size' of the variables. This is known as the domain and range of a function. These two terms are explained below.

Note that $\mathrm{ℝ}$ above represents the set of all real values that represents the interval $(-\infty ,\infty )$. Here is a visual representation of a domain and range with regards to a function. Recall the example of the function f we had introduced earlier.

Graphical representation of a domain, range and function, StudySmarter Originals

This representation suggests that a function works like a machine that transforms elements of the domain, the inputs, into elements of the codomain. The actual outputs of this "transformation machine" will be the elements of the range, the outputs.

There are many types of functions to consider in the realm of mathematics. We have polynomial functions, exponential functions, trigonometric functions, etc. In the following sections, we shall summarize the general formulas used to find the associated domain and range of each type of function (usually seen in this syllabus).

Domain and Range of Polynomial Functions

We shall first note the three types of polynomials we shall often use throughout this topic.

1. linear functions, $f\left(x\right)=ax+b$

2. quadratic functions, $f\left(x\right)=a{x}^2+bx+c$

3. cubic functions, $f\left(x\right)=a{x}^3+b{x}^2+cx+d$

The domain of any polynomial function is the set of all real numbers, IR.

The range of a linear and cubic function is also the set of all real numbers, IR.

The range of a quadratic function of the form$f\left(x\right)=a{(x-h)}^2+k$is

$Range\left(f\right)=\left\{f\right(x)\ge k,ifa>0\}$ or $Range\left(f\right)=\left\{f\right(x)\le k,ifa<0\}$.

Domain and Range of Square Root Functions

For the standard square root function, $f\left(x\right)=\sqrt{x}$, the domain is the set of all real numbers, IR and the range is f(x) ≥ 0.

For a general square root function of the form $f\left(x\right)=\sqrt{g\left(x\right)}$, where g(x) is a function of x, the domain is the set of functions where g(x) ≥ 0 and the range is f(x) ≥ 0.

Domain and Range of Cube Root Functions

For any function containing a cube root, may it be the standard form $f\left(x\right)=\sqrt[3]{x}$ or the general form $f\left(x\right)=\sqrt[3]{g\left(x\right)}$, the domain and range are both the set of all real numbers, IR.

Domain and Range of Exponential Functions

For an exponential function of the form $f\left(x\right)=a^x$, where a is any real number, the domain is the set of all real numbers, IR.

The range will always yield positive real values, that is, f(x) > 0.

Domain and Range of Logarithmic Functions

For a logarithmic function of the form $f\left(x\right)={\log }_{a}x$, where a is any real number, the domain is x > 0 while the range is the set of all real numbers.

Domain and Range of Rational Functions

The domain is the set of all real numbers except for which the denominator is equal to zero, that is $\{x\in \mathrm{ℝ}|q\left(x\right)\ne 0\}$.

The range here is the same as the domain of the inverse of this rational function f or in other words, $Range\left(f\right)=Domain\left(f^{-1}\right)$.

Domain and Range of Trigonometric Functions

Observe the graph of the sine (green line) and cosine (blue line) functions, f(x) = sin(x) and f(x) = cos(x), below.

Sine and cosine graph, StudySmarter Originals

Notice that the value of the functions oscillates between –1 and 1 and it is defined for all real numbers. Thus, for each sine and cosine function, the domain is the set of all real numbers, R and the range is –1 ≤ f(x) ≤ 1. The range here can also be denoted by [–1, 1].

Even and Odd Functions

Even and odd functions are functions that satisfy a particular rule of symmetry. To check whether a function is even or odd, all we need to do is substitute x into the given function and observe whether it satisfies the condition of an even or odd function, which we shall establish below. We shall look at both of these types of functions and identify their respective properties.

Even Function

Geometrically speaking, the graph of an even function is symmetric with respect to the y-axis, in other words, the function remains unchanged when reflected about the y-axis. The properties of even functions include:

• The sum of two even functions is even;

• The difference between two even functions is even;

• The product of two even functions is even;

• The quotient of two even functions is even;

• The derivative of an even function is odd;

• The composition of two even functions is even;

• The composition of an even function and odd function is even.

Let us look at an example.

Odd Function

To look at this geometrically, the graph of an odd function has rotational symmetry with respect to the origin. Essentially, the function remains unchanged when rotated 180^o about the origin. Below are the properties of odd functions:

• The sum of two odd functions is odd;

• The difference between two odd functions is odd;

• The product of two odd functions is even;

• The product of an even function and odd function is odd;

• The quotient of two odd functions is even;

• The quotient of an even function and odd function is odd;

• The derivative of an odd function is even;

• The composition of two odd functions is odd.

Below is an example.

Can a Function be Both Even and Odd?

There is only one function that fulfills this criterion, which is the constant function that is identically zero, f(x) = 0. The domain and range are the set of all real numbers, IR.

It is also possible that we have functions that are neither even nor odd. Here is an example that shows this.

Periodic Functions

Periodic functions are used to describe trigonometric functions in particular due to the presence of oscillations and waves in their graphs.

A function that is not periodic is termed aperiodic. Here is an example of that type of function.

Intercept Points

The points of interception of a function are the points at which the function crosses the axes of the graph. Below is an explicit example of the two points of interception to consider when graphing functions in two dimensions.

Intercept points are important in deducing the change of sign of the curve for a given function. Let us look at an example.

Intersecting Points

Say we are given a pair of functions, f and g. We are told to find the point(s) at which the two functions meet. This is called the intersecting point. This is defined below.

The exact value(s) of the intersection points can be found by solving the expression above algebraically. Below is an example that shows this.

Other Components for Graph Sketching

So far, we have looked at the basic information needed for sketching the graph of a given function. In the following topics in this section, we shall be introduced to other fundamental elements that may be helpful when graphing functions. This includes:

• Finding limits of a function

• Identifying asymptotes. This is explained in the topic of Asymptotes

• Locating the maximum and minimum points of a curve. This can be found here: Maxima and Minima

• Using derivatives to find critical points and flex points. A detailed overview of this topic can be found here: Finding Maxima and Minima using Derivatives

By familiarizing ourselves with these methods, sketching graphs can be much more straightforward and accurate.