Section 1.2 Functions
Subsection 1.2.1 Things to know
A function \(f\) is a rule that assigns to each element \(x\) in a set \(D\) exactly one element, called \(f(x)\text{,}\) in a set \(E\text{.}\)
- 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
The following review videos may be useful: