2 More Functions and Graphs

2.1 Functions

Informal definition Let $A$ and $B$ be sets. A function from $A$ to $B$, denoted by $f:A\rightarrowB$, is a rule that assigns to each element of $A$ exactly one element of $B$.

Terminology The sets $A$ and $B$ are called the domain and codomain of $f$, respectively. The domain is denoted by $\text{dom}(f)$.

Terminology For each $x$ in $A$, the corresponding element assigned by $f$ is denoted by $f(x)$ and is called the image of $x$ under $f$.

Terminology The input variable for a function is called an independent variable. The output variable is called a dependent variable because its value depends on the value of the independent variable.

Example $f:\mathbb{R}\rightarrow\mathbb{R},~x\mapsto f(x)=x^2+2$.

2.2 Domains and Ranges

Natural domain The natural domain of $f$ is the set of all real numbers $x$ such that $f(x)$ is defined.

Example Find the (natural) domains of the following functions:

$g(x)=\frac{1}{x-2}$
$h(x)=\sqrt{1+5x}$

Definition Let $f:A\rightarrow B$ be a function and let $S\subseteq A$. The image of $S$ under $f$, denoted by $f[S]$, is the subset of $B$ given by the subset of $B$ consisting of all the images under $f$ of elements in $S$: \[f[S]=\{y\in B:y=f(x)~~\text{for some}~~x\in S\}.\]

Example Let $f:\mathbb{R}\rightarrow\mathbb{R}, x\mapsto f(x)=x^2$. For $S=\{1,2,3\}$, we have $f[S]=\{1,4,9\}$.

Definition Let $f:A\rightarrow B$ be a function. The range of $f$, denoted by $\text{ran}(f)$, is the image of $A$ under $f$, that is, $\text{ran}(f)=f[A]$.

Example Let $f:\mathbb{R}\rightarrow\mathbb{R}, x\mapsto f(x)=x^2+2$. Then $3$ and $2$ belongs to the range of $f$ but $1$ does not belong to the range of $f$.

Steps to find range of function

Put $y=f(x)$.
Solve $x$ in terms of $y$.
The range of $f$ is the set of all real numbers $y$ such that $x$ can be solved.

Example Find the ranges of the following functions:

$f(x)=x^2+2$
$g(x)=\frac{1}{x-2}$
$h(x)=\sqrt{1+5x}$
$l(x)=\frac{2x+1}{x^2+1}$

2.3 Graphs of Equations

Ordered pair of real numbers An ordered pair of real numbers represents a point in the coordinate plane. The set of all ordered pairs $(x_0,y_0)$ is denoted by $\mathbb{R}^2$, i.e., $(x_0,y_0)\in\mathbb{R}^2$. So the plane is also denoted by $\mathbb{R}^2$.

Functions of two variables $f:A\rightarrow\mathbb{R},(x,y)\in A\mapsto f((x,y))\in\mathbb{R}$, where $A\subseteq\mathbb{R}^2$. Note that $f((x,y))$ is normally simplified as $f(x,y)$.

Example $f:\mathbb{R}^2\rightarrow\mathbb{R}, (x,y)\mapsto f(x,y)=x+y^2.$

Graph of a function Let $F$ be a function of two variables $(x,y)\in\mathbb{R}^2\mapsto F(x,y)\in\mathbb{R}$. Consider an equation in the form of $F(x,y)=0$. The set of all ordered pairs $(x,y)$ satisfying $F(x,y)=0$ is called the graph of the equation $F(x,y)=0$, i.e., it is a subset of the plane $\mathbb{R}^2$: \[\{(x,y)\in\mathbb{R}^2:F(x,y)=0\}.\]

Definition An $x$-intercept ($y$-intercept) of the graph of an equation $F(x,y)=0$ is a point where the graph intersects the $x$-axis ($y$-axis).

Example The graph of equation $x^2+y^2=1$ is a circle with two $x$-intercepts $(1,0),(-1,0)$ and two $y$-intercepts $(0,1),(0,-1)$.

Symmetry Consider the graph of the equation $y=x^2$. The graph is a parabola. If $(a,b)$ is a point belonging to the parabola, then $(-a,b)$ also belongs to the parabola since $b=(-a)^2$. We say that the parabola is symmetric about the $y$-axis.

Symmetry In general, a subset $\mathcal{A}$ of the plane $\mathbb{R}^2$ is said to be symmetric about a line $\mathcal{L}$ if the following condition is satisfied: For any point $P$ belonging to $\mathcal{A}$, there is a point $Q$ belonging to $\mathcal{A}$ such that

the line segment $PQ$ is perpendicular to $\mathcal{L}$;
$P$ and $Q$ are equidistant from $\mathcal{L}$.

2.4 Graphs of Functions

Let $f:A\rightarrow \mathbb{R}$ be a function where $A\subseteq \mathbb{R}$. The graph of $f$ is the following subset of $\mathbb{R}^2$: \[\{(x,y)\in\mathbb{R}^2:x\in A~~\text{and}~~y=f(x)\}.\]

Constant functions $f(x)=c$. The domain is $\mathbb{R}$. The range is a singleton $\{c\}$.

Figure 2.1: Constant function
Linear functions $f(x)=ax+b$. The domain is $\mathbb{R}$. The range is $\mathbb{R}$.

Figure 2.2: Linear function
Quadratic functions $f(x)=ax^2+bx+c$. The domain is $\mathbb{R}$. The range is $[k,\infty)$ if $a>0$ and $(-\infty,k]$ if $a<0$, where $k$ is the $y$-coordinate of the vertex.

Figure 2.3: Quadratic function
Polynomial functions with degree $n$ $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. The domain is $\mathbb{R}$. There are three possibilities for the range.
1. If the degree $n$ is odd, then $\text{ran}(f)=\mathbb{R}$.
2. If the degree $n$ is even and positive, then
- If $a_n>0$ $\text{ran}(f)=[k,\infty)$;
- If $a_n<0$ $\text{ran}(f)=(\infty,k]$.

Example $f(x)=x^3-3x^2+x-1.$

Figure 2.4: Polynomial function

Rational functions are in the form of \[f(x)=\frac{p(x)}{q(x)},\] where $p$ and $q$ are polynomial functions.

Example $f(x)=\frac{1}{x}$. The domain of $f$ is $\mathbb{R}\backslash \{0\}$. The range of $f$ is also $\mathbb{R}\backslash \{0\}$. The graph of it is symmetric about the origin because $f(-x)=-f(x)$.

Figure 2.5: Rational function

Principle-square-root function. Denoted by sqrt or $\sqrt{~~}$. Given by $\text{sqrt}(x)=\sqrt{x}$. The domain is $[0,\infty)$. The range is $[0,\infty)$.

Figure 2.6: Square-root function

Example For each of the following equations, sketch its graph.

$y=\sqrt{x}-2$

Figure 2.7: Square-root function
$y=\sqrt{x-2}$

Figure 2.8: Square-root function
$y=\sqrt{2-x}$

Figure 2.9: Square-root function

Remark Let $a$ be a positive constant.

The graph of $y=f(x)+a$ can be obtained from that of $y=f(x)$ by moving it $a$ units up.
The graph of $y=f(x)-a$ can be obtained from that of $y=f(x)$ by moving it $a$ units down
The graph of $y=f(x+a)$ can be obtained from that of $y=f(x)$ by moving it $a$ units to the left.
The graph of $y=f(x-a)$ can be obtained from that of $y=f(x)$ by moving it $a$ units to the right.
The graph of $y=\sqrt{2-x}$ and that of $y=\sqrt{x-2}$ are symmetric w.r.t. the vertical line $x=2$.

Exponential functions with base $b$ is given by \[\exp_b(x)=b^x\]. The domain is $\mathbb{R}$. The range is $(0,\infty)$. The $y$-intercept of the graph of every exponential function is $(0,1)$.

Example The graph of $\exp_2$ goes up and the rate that the graph goes up increases as $x$ increases.

Figure 2.10: Exponential function with base larger than 1

Example The graph of $\exp_{\frac{1}{3}}$ goes down as $x$ increases.

Figure 2.11: Exponential function with base smaller than 1

Logarithmic functions. The function, denoted by $\log$, is called the common logarithmic function. For each positive real number $x$, $\log(x)$ is defined to be the unique real number such that $10^{\log (x)}=x$. That is, $\log(x)=y$ if and only if $y=10^x$. The domain of log is $(0,\infty)$. The range is $\mathbb{R}$.

Figure 2.12: Logarithmic function

Absolute value function, denoted by $|\cdot|$, is the function from $\mathbb{R}$ to $\mathbb{R}$, given by \[\begin{equation} |x|= \begin{cases} x~~&\text{if}~~ x>0,\\ 0~~&\text{if}~~ x=0,\\ -x~~&\text{if}~~x<0. \end{cases} \end{equation}\]

In defining $|x|$, the domain $\mathbb{R}$ is divided into three disjoint subsets, namely $(0,\infty)$, $\{0\}$ and $(-\infty,0)$. Functions defined in this way are called piecewise-defined functions.

Figure 2.13: Absolute value function

Piecewise-defined functions

Example Let $f:[-2,6]\rightarrow\mathbb{R},x\mapsto f(x)$ be the function given by \[\begin{equation} f(x)= \begin{cases} x^2~~&\text{if}~~-2\le x<0\\ 2x~~&\text{if}~~0\le x<2\\ 4-x~~&\text{if}~~2\le x\le 6 \end{cases} \end{equation}\]

Figure 2.14: Piecewise function

Step function Suppose the long-distance rate for a telephone call from City A to City B is $1.4 for the first minute and $0.9 for each additional minute or fraction thereof. If $y=f(t)$ is a function that indicates the total charge $y$ for a call of $t$ minutes’ duration, sketch the graph of $f$ for $0<t\le 4\frac{1}{2}$. \[\begin{equation} f(t)= \begin{cases} 1.4 ~~&\text{if}~~0<t\le 1,\\ 2.3 ~~&\text{if}~~1<t\le 2,\\ 3.2 ~~&\text{if}~~2<t\le 3,\\ 4.1 ~~&\text{if}~~3<t\le 4,\\ 5.0 ~~&\text{if}~~4<t\le 4\frac{1}{2}. \end{cases} \end{equation}\]

Figure 2.15: Step function

2.5 Compositions of Functions

Definition Let $f$ and $g$ be functions such that the codomain of $f$ is a subset of the domain of $g$. The composition of $g$ with $f$, denoted by $(g\circ f)(x)=g(f(x)).$

Example Let $f(x)=x^2$ and $g(x)=2x+1$. Find $(f\circ g)(x)$ and $(g\circ f)(x)$.

Remark The composition of functions is not commutative.

Remark If the range of $f$ is not contained in the domain of $g$, then we have to restrict $f$ to a smaller set so that for every $x$ in that set, $f(x)$ belongs to the domain of $g$. The domain of $g\circ f$ is taken to be the following: \[\text{dom}(g\circ f)=\{x\in\text{dom}(f):f(x)\in\text{dom}(g)\}.\]

Example Let $f(x)=x+1$ and $g(x)=\sqrt{x}$. Find the domain of $g\circ f$.

2.6 Inverse Functions

Definition Let $f$ be a function. We say that $f$ is injective if the following condition is satisfied: \[x_1,x_2\in\text{dom}(f)~~\text{and}~~ x_1\neq x_2 \implies f(x_1)\neq f(x_2)\] which is also equivalent to \[x_1,x_2\in\text{dom}(f)~~\text{and}~~ f(x_1)=f(x_2) \implies x_1=x_2.\]

Example Let $g(x)=x^2$. The domain of $g$ is $\mathbb{R}$. The function $g$ is not injective.

Horizontal line test Let $f: X\rightarrow \mathbb{R}$ be a function where $X\subseteq\mathbb{R}$. Then $f$ is injective if and only if every horizontal line intersects the graph of $f$ in at most one point.

Figure 2.16: Horizontal line test

Definition Let $f: X\rightarrow Y$ be an injective function and let $Y_1$ be the range of $f$. The inverse function of $f$, denoted by $f^{-1}$, is the function from $Y_1$ to $X$ such that for every $y\in Y_1$, $f^{-1}(y)$ is the unique element of $X$ satisfying $f(f^{-1}(y))=y$.

Figure 2.17: Injective mapping

Remark

For every $x\in X$, we have $(f^{-1}\circ f)(x)=x.$ For every $y\in Y$, we have $(f\circ f^{-1})(y)=y$.
$f^{-1}$ is injective and $(f^{-1})^{-1}(x)=f(x)$ for all $x\in\text{dom}(f).$

Steps to find inverse functions

Put $y=f(x)$.
Solve $x$ in terms of $y$. The result will be in the form $x=$ an expression in $y$.
From the expression in $y$ obtained in step 2, the range of $f$ can be determined. This is the domain of $f^{-1}$. The required formula $f^{-1}(y)=$ the expression in $y$ obtained in Step 2.

Example Let $f(x)=2x^3+1$. Find the inverse of $f$.

Example Let $g: [0,\infty)\rightarrow \mathbb{R}, x\mapsto g(x)=x^2$. Find the inverse of $g$.

Graph of the inverse function The graph of $f$ and the graph of $f^{-1}$ are symmetric about the line $y=x$.

Figure 2.18: Inverse function

2.7 More on Solving Equations

In this section, we will consider fractional equations and radical equations. In solving equations, if there is a one-sided implication $\implies$ in any one of the step, we have to check solution.

Example For the following equations, find their solution sets, respectively.

$\frac{x}{x-1}+\frac{2}{x}=\frac{1}{x^2-x}$
$\sqrt{x}-\sqrt{x-3}=3$

Remark

$a=b\implies ac=bc$, but the converse is true only if $c\neq 0$.
$a=b \implies a^2=b^2$, but the converse is true only if $a$ and $b$ have the same sign.