Functions

Math 152

Definition

• A function is a rule that associates each element of a set X, called the domain with a unique element of another set Y, called the codomain. The range f is the set of all elements in Y which correspond to an element of X

Domain $x\in \mathbb{R}$ is all number Co-domain is possible integers from the function Range is is the direct output of x

In calculus we will only consider fuctions whose domain and codomain consist of real numbers scribed in various ways

1. Verbally

• ex. The area of a circle is $\pi$ times

2.

Example

• 1. Function evaluation:

  • $f(x)=(x+2{)}^2-x$
    • ⇒$f(a+h)=((a+h)+2{)}^2-(a+h)$

• 2. finding the domain: what cant I do

  • no /0
  • no even roots of negative numbers
  • no nonpositive logs

• 3. Finding the range

  • look at transformation of or for inverse

• 4. graphing a function

  • set of points $(x,f(x)=y$)

• 5. Identifying even/ odd symmetry

  • Examining f(-x) →
    • $=f(x)?$ yes even →
    • = -f(x)? If yes odd

• 6. Identifying increasing/decreasing intervals

  • calcsus

Function Notation

• Linear Functions

  • $f(x)=mx+b$

• Power Functions

  • $f(x)=x^a$

• polynomial functions

  • $f(x)=a{x}^n+a{x}^{n-1}$

• algebraic Functions

  • combined: algebraic (including roots)

• Trigonometric Functions

Transformations

• Vertical Translation

  • $f(x)=(x)+c,(c=\uparrow )$
  • $f(x)=(x)-c,(c=\downarrow )$

• Horizontal Translation

  • $f(x)=(x-c),(c=\to )$
  • $f(x)=(x+c),(c=\leftarrow )$

• Vertical Stretch

  • $(a,cf(x)\to ,(c=\updownarrow ))$

• Horizontal Stretch

  • $(ca,f(x)\to ,(c=\leftrightarrow ))$