Download Notes 4 - Garnet Valley School District

Notes 4.1 & 4.5 – Functions and Inverses
Pre-Calc. for AP Prep.
Date: _________
Goal:
We have been using function notation since the start of the course, but what exactly is a
function?
A function ______________________________________________________________
_______________________________________________________________________
Set D
Set R
The set D refers to the domain of the function, or the possible values of _____________.
The set R refers to the range of the function, or the possible values of ______________.
Since a function is mapping every X to exactly one Y, functions pass the vertical line test.
The vertical line test states ________________________________________________
________________________________________________________________________
Examples:
Determine if the graphs/equations shown are functions. Then find the domain and range.
1)
2)

y





x











x = y2
We can perform a variety of operations on functions. They are as follows:
1.
(f + g)(x) = f(x) + g(x)
2.
(f – g)(x) = f(x) – g(x)
3.
(f  g)(x) = f(x)  g(x)
f 
f ( x)
4.
assuming g(x)  0
  ( x) 
g ( x)
g
5.
 f g  ( x)  f ( g ( x))
Examples:
Let f(x) = x4 – x2, m(x) = x – 1 and j(x) = 2 x 2  3 x . Find the following:
1)
m
   x
 f 
2)
 j  mx
j x  h   j ( x)
h
3)
II. Inverses of Functions
1. Formal Definition Two functions, f and g are inverse functions if two conditions are
met.
a. f(g(x)) = x for all x in the domain of f.
b. g(f(x)) = x for all x in the domain of g.
2. Informal Definition – Inverses function “undo” each other.
Example:
a.
Find the inverse function “ f
b.
Find f
1
x  .
c.
1
x  ” given
Find
f ( x)  3  x .
 x
f f
1
d.
Find f 1  f ( x)
Inverses which are functions, the horizontal line test and domain restrictions.
1. Recall that for a relation to be a function,
it must pass the vertical line test.
2. Also, recall that to graph the inverse of a function, we can translate each point
(x, y) to (y, x). Generally, the resulting graph is a reflection of the original over
the line y = x.
Sketch a graph of the inverse of y = x2.
3. To discuss the inverse relation as an inverse function is technically incorrect.
4. Depending on the application, we can either say…
a. Since the original function fails the horizontal line test, it does not
have an inverse function.
or
b. Define F(x), the original function with a limited domain that would
pass the horizontal line test and therefore has an inverse function.
Ex: Give a domain limitation for f ( x)  x 2 that allows f
exist.
1
( x) to
Check for Understanding
1. Suppose f is a function with an inverse. If f(-1) = 5 and f(3) = -2, find the following:
a.
f-1(-2)
= ____________
b. f(f-1(5))
= ___________
HW: p. 112 11, 12; p. 123 2, 4, 6, 10, 12, 17; p. 149 2, 3, 8, 10, 14, 18, 22