1.1 Functions and their graphs

本文爲《Thomas’ Calculus Early Transcendentals》閱讀筆記

1. Definition: A function $f$ from a set $D$ to a set $Y$ is a rule
   that assigns a unique(single) element $f(x) \in Y$ to each elements
   $x \in D$
2. The set $D$ of all possible input values is called the domain of the function.
3. The set of all output values of $f(x)$ as $x$ varies throughout $D$ is called the range of the function
4. If $f$ is a function with domain $D$, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for $f$.
   
5. Another way to represented function is numerically, through a table of values. The graph consisting of only the points in the table is called a scatterplot.
   
6. Not every curve of coordinate plane can be the graph of a function. A function $f$ can have only one value $f(x)$ for each $x$ in the domain, so no vertical line can intersect the graph of a function more than once. If $a$ is in the domain of function $f$, then the vertical line $x = a$ will intersect the graph of $f$ at the single point $(a, f(a))$
7. Sometimes a function is described in pieces by using different formulas on different parts of its domain. One example is the absolute value function:
   $|x| = \begin{dcases} x,&x \ge 0\\ -x,&x < 0 \end{dcases}$
8. Definition: Let $f$ be a function defined on an interval $I$ and let $x_{1}$ and $x_{2}$ be any points in $I$.
        1) If $f(x_{2}) > f(x_{1})$ whenever $x_{1} < x_{2}$, then $f$ is said to be increasing on $I$
        2) If $f(x_{2}) < f(x_{1})$ whenever $x_{1} < x_{2}$, then $f$ is said to be decreasing on $I$
9. Definition: A function $y = f(x)$ is an
         even function of x if $f(-x) = f(x)$
         odd function of x   if $f(-x) = -f(x)$
   for every $x$ in the function’s domain.
   The graph of an even function is symmetric about the y-axis.
   The graph of an odd function is symmetric about the origin.
10. A function of the form $f(x) = mx + b$, for constants $m$ and $b$, is called a linear function.
11. A function $f(x) = x^a$, where $a$ is a constant, is called a power function.
12. A function $p$ is a polynomial if
    $p(x) = a_{n} x^n + a_{n-1} x^{n-1} + \ldots + a_{1}x + a_{0}$
    where $n$ is a nonnegative integer and the numbers $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ are real constants(called the coefficients of the polynomial).
13. A rational function is a quotient or ratio $f(x) = p(x)/q(x)$, where $p$ and $q$ are polynomials. The domain of a rational function is the set of all real $x$ for which $q(x) \neq 0$.
14. Any function constructed from polynomials using algebraic operations (addition, subtraction, division, multiplication, and taking roots) lies within the class of algebraic functions.
15. The six basic trigonometric functions are reviewed in Section 1.3.
16. Functions of the form $f(x) = a^{x}$, where the base $a > 0$ is a positive constant and $a \neq 1$, are called exponential function.
17. Logarithmic functions are the functions $f(x) = \log_{a}x$, where the base $a \neq 1$ is a positive constant.

Exercises
In exercises 1-6, find the domain and range of each function.

1. $f(x) = 1 + x^2$ domain: $x \in (-\infty, +\infty)$ range: $f(x) \in [1, +\infty)$
2. $f(x) = 1 - \sqrt{x}$ domain: $x \in [0, +\infty)$ range: $f(x) \in (-\infty, 1]$
3. $F(x) = \sqrt{5x + 10}$ domain: $x \in [-2, +\infty)$ range: $F(x) \in [0, +\infty)$
4. $g(x) = \sqrt{x^2 - 3x}$ domain: $x \in (-\infty, 0] \bigcup [3, +\infty)$ range: $g(x) \in [0, +\infty)$
5. $f(t) = \frac{4}{3-t}$ domain: $t \in (-\infty,3) \bigcup (3, +\infty)$ range: $f(t) \in (-\infty, +\infty)$
6. $G(t) = \frac{2}{t^2 -16}$ domain: $t \in (-\infty,4) \bigcup (4, +\infty)$ range: $G(t) \in (-\infty, -\frac{1}{8}] \bigcup (0, +\infty)$