2.10 Examples of Elementary Functions

You should be familiar with the graphs of some functions that frequently occur in applications.

Constant functions

A function which for all values of $x$ in an interval has the same value $y=c$ is called a constant function. The graph of a constant function is a horizontal straight line.

Linear functions

Functions of the following form are called linear functions:
\[
f(x)=mx+b
\] where $m$ and $b$ are fixed constants. Figure 1. shows the general graph of linear functions.

• $m$ is the slope of the line. [That is, if the angle between the line and the $x$-axis is denoted by $\theta$, then $m=\tan\theta$.We will explain this further in the next chapter]
• $y=b$ is where the line intersects the $y$-axis.
• If $m=0$, we have a constant function $f(x)=b$.
• If $b=0$, the line passes through the origin.
• We say two variables $y$ and $x$ are proportional to one another if there exists a nonzero constant $k$ such that $y=kx$.
• We say two variables $y$ and $x$ are inversely proportional to one another if there exists a nonzero constant $k$ such that $y=\dfrac{k}{x}.$

Power functions

A function of the form

\[
f(x)=x^{a},
\] where $a$ is a constant is called a power function. There are some important cases that we need to consider.

(a) When $a=n$ is a positive integer.

The graph of $y=x^{n}$is of two distinct types, depending whether $n$ is even or odd.

• When $n$ is even:
  • The general graph of $y=x^{n}$ ($n$ is even) is shown in Figure 2(a).
  • As we can see the graph will rise without bound as $x$ moves to the right or left.
  • The function is even because
    \[
    f(-x)=(-x)^{n}=x^{n}=f(x),
    \] and hence the graph of $y=x^{n}$ is symmetric about the $y$-axis.
  • The range of $y=x^{n}$ ($n$ is even) is $[0,\infty)$.
  • The graphs of $y=x^{2}$, $y=x^{4}$, and $y=x^{6}$ are compared
    in Figure 2(b). In general
    \[
    x^{2}>x^{4}>x^{6}>x^{8}>\cdots\quad\text{when }-1<x<1
    \] and
    \[
    x^{2}<x^{4}<x^{6}<x^{8}<\cdots\quad\text{when }x>1\text{ or }x<-1.
    \]
  • The larger the exponent $n$, the steeper the graph becomes for $x<-1$ or $x>1$, and the flatter and closer to the $x$-axis the graph is for $x$ near the origin (see Figure 2(b)).

(a) Typical graph of $ n $ is even.	(b) A family of power functions with even exponents

Figure 2

• When $n$ is odd:
  • The general graph of $y=x^{n}$ when $n$ is odd is shown in Figure3(a).
  • The function is odd too because
    \[
    f(-x)=(-x)^{n}=-x^{n}=-f(x).
    \] Thus, the graph of $y=x^{n}$ ($n$ is odd) is symmetric about the origin $(0,0)$.
  • As we can see, the graph will rise without bound as $x$ moves to the right and will fall without bound as $x$ moves to the left.
  • The range of $y=x^{n}$ (for odd $n$) is $\mathbb{R}=(-\infty,\infty)$.
  • The graphs of $y=x$, $y=x^{3}$, and $y=x^{5}$ are compared in Figure3(b). In general
    \[
    |x|>|x^{3}|>|x^{5}|>|x^{7}|>\cdots\quad\text{when }-1<x<1
    \] and
    \[
    |x|<|x^{3}|<|x^{5}|<|x^{7}|<\cdots\quad\text{when }x>1\text{ or }x<-1
    \]
  • The larger the exponent $n$, the steeper the graph becomes for $x<-1$ or $x>1$, and the flatter and closer to the $x$-axis the graph is for $x$ near the origin (see Figure 3(b)).

(a): Typical graph of $y=x^{n} $when $n $ is odd.	(b) A family of power functions with odd exponents

Figure 3

(b) When $a=-1$ or $a=-2$.

• The function $f(x)=1/x$ :
  • The graphs of $y=1/x$ is shown in Figure 4(a).
  • The function is odd:
    \[
    f(-x)=\frac{1}{-x}=-\frac{1}{x}=-f(x).
    \] Hence, its graph is symmetric about the origin.

• • The function is defined for all values of $x\neq0$ (division by zero is not defined).
  • The absolute value of $y$ becomes very small for very large (positive or negative) values of $x$.
  • As $x$ increases to 0 (i.e., $x$ approaches 0 from the left), $y$ is negative and the graph will fall without bound; while as $x$ decreases to 0 (i.e. $x$ approaches 0 from the right), $y$ is positive and the graph will rise.
  • The range of $f(x)=1/x$ is the entire set of real numbers except $y=0$:
    \[
    Rng(f)=\{y|\ y\neq0\}
    \]

• The function $f(x)=1/x^{2}$:
  • The graphs of $y=1/x^{2}$ is shown in Figure4(b).
  • The function is even,
    \[
    f(-x)=\frac{1}{(-x)^{2}}=\frac{1}{x^{2}}=f(x)
    \] Hence, its graph is symmetric about the $y$-axis.
  • The function is defined for all values of $x\neq0$ (division by zero is not defined).
  • The value of $y$ becomes very small for very large (positive or negative) values of $x$.
  • The graph will rise without bound as $x$ increases or decreases to 0 (i.e. as $x$ approaches 0 either from the right or left).
  • The range of $f(x)=1/x^{2}$ is the set of positive numbers $(0,\infty)$:
    \[
    Rng(f)=\{y|\ y>0\}=(0,\infty).
    \]

(c)When $a=1/2,a=1/3$.

The functions\[
y=x^{1/2}=\sqrt{x}\qquad\text{or}\qquad y=x^{1/3}=\sqrt[3]{x}.
\] are square root and cubic root, respectively. The graphs of these functions are shown in Figure 5.

• Because $\sqrt{x}$ is defined only for nonnegative values of $x$,the domain of the square root function is thus $[0,\infty)$.
• The cubic root $\sqrt[3]{x}$ is defined for all $x$, thus the domain of $y=\sqrt[3]{x}$ is $\mathbb{R}$.
•  As we can see the range of $y=\sqrt{x}$ is $[0,\infty)$ and the range of $y=\sqrt[3]{x}$ is the entire set of real numbers $\mathbb{R}$.
•  Note that $y=\sqrt[3]{x}$ is an odd function but $y=\sqrt{x}$ is neither odd nor even because it is not defined for the negative values of $x$.

Polynomial functions

A polynomial in $x$ is a function of the form
\[
f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0},
\] where the exponents of $x$ are nonnegative integers. The numbers $a_{0},\cdots,a_{n}$ are called the coefficients of the polynomial. If $a_{n}\neq0$, we say $f$ is a polynomial of degree $n$.

• To emphasize that the function is a polynomial of degree $n$, many authors use $P_{n}(x)$ instead of $f(x)$.
• Constant functions and the power functions $y=x^{m}$ where $m$ is a nonnegative integer are special cases of polynomials.
• Polynomials of degree 1, 2, 3, and 4 are called linear, quadratic, cubic, and quartic polynomials, respectively: If $a\neq0$,

• The graph of a polynomial
  \[
  f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0},
  \] may have several turns but its graph eventually will rise or fall boundlessly as $x$ moves to the right or left depending on the dominant term $a_{n}x^{n}$:
• If $n$ is even and $a_{n}>0$, the graph will rise (Figure 6(a)) and if $a_{n}<0$ the graph will fall (Figure 6(b)).

(a) $n$ is even and $a_n>0$	(b) $n$ is even and $a_n<0$

Figure 6

• If $n$ is odd and $a_{n}>0$ the graph will rise on the right side and will fall on the left side (Figure 7(a)) and if $a_{n}<0$ the graph will fall on the right side and will rise on the left side of the graph (Figure 7(b)).

(a) $n$ is odd and $a_n>0$	(b) $n$ is odd and $a_n<0$

Figure 7

Rational functions

A rational function $f(x)$ is a function that can be written as the quotient of two polynomials, namely
\[
f(x)=\frac{a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}}{b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}}.
\] For example
\[
f(x)=\frac{5x^{3}+4x+1}{-x^{5}+x^{2}+3}\qquad\text{or}\qquad f(x)=\frac{-x^{4}+5}{x^{2}-3x+2},
\] are rational functions.

To determine the domain of a rational function $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, we have to exclude the roots of $Q(x)$; that is, the domain of $f$ is the entire set of
real numbers excluding the values of $x$ for which $Q(x)=0$:
\[
Dom(f)=\{x|\ Q(x)\neq0\}.
\]

Irrational functions

If a function requires the use of radical signs combined with polynomials, it is an example of an irrational algebraic function; for example,

\[
f(x)=\sqrt[3]{x^{3}+\sqrt{\frac{x}{1+x^{2}}}}.
\]

Algebraic functions

The functions that are generated by a finite number of addition, subtraction, multiplication, division, and raising to a fractional power are called algebraic functions. Note that polynomials, rational, and irrational algebraic functions are algebraic functions.

Transcendental functions

Any function of $x$ which is not algebraic is called transcendental. The elementary transcendental functions are the trigonometric, the inverse trigonometric, the exponential, and the logarithmic functions, the definitions and the simplest properties of which are supposed to be known to the students. We will study elementary transcendental functions in the next chapter.