MATH 144: Calculus for the Physical Sciences I

Subsection 1.2.1 Things to know

• Domain and range: The set $D$ is called the domain of $f\text{,}$ while the range of $f$ is the set of all possible values of $f(x)$ as $x$ varies through the domain.
• Graph of a function: all points in the $xy$-plane such that $y=f(x)$ with $x$ in the domain of $f\text{.}$
• Vertical line test: A curve in the $xy$-plane is the graph of a function of $x$ if and only if no vertical line intersects the curve more than once.
• Piecewise defined functions: Functions that are defined using different formulae for different parts of their domains.
• Absolute value function: $f(x) = |x|$ is defined by
  $$\begin{equation*} |x| = \begin{cases} x \amp \text{for $x\geq 0$},\\ -x \amp \text{for $x\lt 0$}. \end{cases}. \end{equation*}$$
• Increasing and decreasing functions: A function is increasing (resp. decreasing) on some interval $I$ if $f(x_1)\lt f(x_2)$ (resp. $f(x_1) \gt f(x_2)$) for all $x_1, x_2 \in I$ such that $x_1 \lt x_2$ (resp. $x_1 \gt x_2$).
• Odd and even functions: A function $f$ such that $f(x) = f(-x)$ is even, while if $f(x) = - f(-x)$ it is odd.
• Transformations of functions:
  • Vertical shift: $y = f(x) + k\text{,}$
  • Horizontal shift: $y = f(x -k)\text{,}$
  • Vertical stretch: $y = c f(x)\text{,}$
  • Horizontal stretch: $y = f(x/c)\text{,}$
  • Reflection about the $x$-axis: $y = - f(x)\text{,}$
  • Reflection about the $y$-axis: $y = f(-x)\text{,}$
  • Composition of functions: $(f \circ g)(x) = f(g(x))\text{.}$ Note that the domain of $f \circ g$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f\text{.}$
• Types of functions:
  • Linear function: $f(x) = mx + b\text{.}$ Its graph is a line with slope $m$ and $y$-intercept $b\text{.}$
  • Polynomial function: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\text{.}$ The degree of $f(x)$ is $n\text{.}$ When $n=2\text{,}$ $f(x)$ is called quadratic, while it is called cubic if $n=3\text{.}$
  • Power function: $f(x) = x^a\text{.}$ If $a$ is a positive integer, then $f(x)$ is a particular example of a polynomial function. If $a=1/n$ with $n$ a positive integer, then $f(x)$ is a root function (for example, for $a=1/2$ it is the familiar square root function $f(x) = x^{1/2} = \sqrt{x}$). For $a=-1\text{,}$ it is the reciprocal function $f(x) = 1/x\text{.}$
  • Rational function: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomial functions.
  • Algebraic function: A function that can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with a polynomial function. All rational functions are clearly algebraic.

Subsection 1.2.2 Review videos