Functions f(x)

Functions f(x)

1. Characteristics of a Function

   • each element of set A must be matched with an element of B
   • 2) some elements of B may not be matched with any element A
   • 3) 2 or more elements of A may be matched with the same element of B
   • 4)an element of A cannot be matched with two elements of B
   • SUMMARY Each element of A (Domain or inputs) must be matched with one and only one element of B (range or outputs)
     • use this to test if a relation is a function
       • vertical line test is used for a graph( for each x there is only one y
       • for each input there must be one and only one output

2. Domain: all values of x that make the function f(x) defined. Range:The range of a function is the set of all values(outputs) assumed by the dependent variable

3. 4 ways to present a function

   • verbally describe the relationship between the independent variable x and dependent variable y or f(x)(ex: year 2001 the profit is 2 billions dollars, year 2004 the profit is 4 billions dollars...)
   • use the graph with x-coordinate and y-coordinate to show the relationship between x & f(x)
   • numerically using the table or set of data to show the relationship (ex. (2001, 2billions dollars), (2004, 4 billions dollars))
   • algebraically using the equation to show the relationship between x & f(x) (ex: f(x) = x+5

4. even and odd functions

   • even functions: when each x in the domain of f, f(-x)=f(x)
     • graphically reflected in the y- axis
   • odd functions: when each x in the domain of f, f(-x)=-f(x)
     • reflected in the origin

5. Combination of functions

   • sum: (f+g)(x)= f(x)+g(x)
   • difference: (f-g)(x)= f(x)-g(x)
   • product: (fg)(x)= f(x)*g(x)
   • quotient: (f/g)(x)= f(x)/g(x), g(x)does not =0
   • the domain is the combination of all domains
   • if f(x) and g(x) are both odd or even functions then their product is an even function (pg 59 problem 93)
     • if f(x) and g(x) are NOT both odd or even functions then their product is an odd function (pg 59 problem 94)
   • The domain of the combinations of f(x)and g(x) = D of f(x) AND D of g(x)

6. Composition of Functions(nesting): f(g(x)), ex. f(x)=x^2, g(x)= x+1

   • f(g(x))= f(x+1)= (x+1)^2
   • the domain is the combination of all domains
   • The domain of the compositions of f(x)and g(x) : 1) D of (f(g(x))= D of f(g(x)) AND D of g(x) 2) D of (g(f(x))= D of g(f(x)) AND D of f(x)

7. Inverse functions f(x), g(x)

   • how to verify f(x) and g(x) are inverse functions of each other (3 ways)
     • identity function (algebraically) test the identity function,f(g(x))= x AND g(f(x))=x
     • numerically (graphing utility, table) Domain of f(x) = range of g(x) and Domain of g(x) = range of f(x)
     • graphically: the graph of f(x) and g(x) are reflected over the y=x line
   • how to find an inverse function algebraically (5 steps)
     • 2) in the equation, replace f(x) by y
     • 3) interchange the roles of x and y, and solve for y
     • 4) replace y by f^-1(x) in the new equation
     • 1) use the horizontal line test to decide whether f(x) has an inverse function or not
     • 5) verify that f and f^-1 are inverse functions of each other by showing that the domain of f is the range of f^-1, the range of f is equal to the domain of f^-1, and f(f^-1(x))and f^-1(f(x))= x (identity function)
   • condition for a function f(x) to have an inverse function (condition for the existence of an inverse function)
     • f(x) must be a one-to-one function
       • definition of one-to-one function
         • no two elements in the domain correspond to an element in the range, f(a) = f(b) implies a = b
       • 2 special types of functions that pass the hozisontal test
         • If f(x) is increasing on its entire domain
           • ex) y=mx+b, m is a positive number, or sin(x) with domain [-pi/2, pi/2]
         • if f(x) is decreasing on its entire domain
           • ex) y=mx+b, m is a negative slope
       • tests for one-to-one function
         • graphically : horizontal line test
         • algebraically: f(a) = f(b) implies a = b
   • the graph of an inverse function
     • the graph is a reflection in the line y=x
   • the identity function
     • f(f^-1(x))= x, f^-1(f(x))=x
   • if f(x) and g(x) are inverse of each other , then
     • enter the graphs:y1= f(x), y2=g(x), y3=f(g(x)), y4=g(f(x)) tab;e feature shows x=y3=y4 means f(g(x) = x and g(f(x)) = x

8. find Domain

   • even root : the expression inside the even root must >=0, ex square rppt of (1-x) , (1-x)>=0, x<=1
   • fraction: the expression in teh denominator must NOt equal to zero, ex: 1/(1-x), (1-x) !=0, x!=1