1.1 Functions and their graphs
阿新 • • 發佈:2019-02-07
本文為《Thomas’ Calculus Early Transcendentals》閱讀筆記
- Definition: A function from a set to a set is a rule that assigns a unique(single) element to each elements
- The set of all possible input values is called the domain of the function.
- The set of all output values of as varies throughout is called the range of the function
- If is a function with domain , its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for .
- 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
- Not every curve of coordinate plane can be the graph of a function. A function can have only one value for each in the domain, so no vertical line can intersect the graph of a function more than once. If is in the domain of function , then the vertical line will intersect the graph of at the single point
- Sometimes a function is described in pieces by using different formulas on different parts of its domain. One example is the absolute value function:
- Definition: Let be a function defined on an interval and let and be any points in .
1) If whenever , then is said to be increasing on
2) If whenever , then is said to be decreasing on - Definition: A function is an
even function of x if
odd function of x if
for every 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. - A function of the form , for constants and , is called a linear function.
- A function , where is a constant, is called a power function.
- A function is a polynomial if
where is a nonnegative integer and the numbers are real constants(called the coefficients of the polynomial). - A rational function is a quotient or ratio , where and are polynomials. The domain of a rational function is the set of all real for which .
- Any function constructed from polynomials using algebraic operations (addition, subtraction, division, multiplication, and taking roots) lies within the class of algebraic functions.
- The six basic trigonometric functions are reviewed in Section 1.3.
- Functions of the form , where the base is a positive constant and , are called exponential function.
- Logarithmic functions are the functions , where the base is a positive constant.
Exercises
In exercises 1-6, find the domain and range of each function.
- domain: range:
- domain: range:
- domain: range:
- domain: range:
- domain: range:
- domain: range: