Building Functions from Functions

1
Building Functions from Functions

• 1.4

2
Sum, Difference, Product, and Quotient of
Functions

• Let f and g be two functions with intersecting
  domains. Then for all values of x in the
  intersection, the algebraic combinations of f and
  g are defined by the following rules

3
Sum, Difference, Product and Quotient

• Sum (fg)(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), provided g(x)?0.

4
Sum, Difference, Product and Quotient

• In each case, the domain of the new function
  consists of all numbers that belong to both the
  domain of f and the domain of g. As noted, the
  zeros of the denominator are excluded from the
  domain of the quotient.

5
Find fg, f-g, fg. Give the domain of each
function and the sum, difference and product.

• Demonstration in Geometer Sketchpad.
• Examples 2,4,6

6
Composition of Functions

• Let f and g be two functions such that the domain
  of f intersects the range of g. The composition
  f of g, denoted f º g, is defined by the rule
• (fºg)(x)f(g(x))
• The domain of f º g consists of all x-values in
  the domain of g that map to g(x) values in the
  domain of f.

7
Composition of Functions

• Find the domain of the inside function
• Find the domain of the outside function.
• The domain of the composition is the domain of
  the inside function with numbers taken out that
  produce values that are not in the domain of the
  outside function.

8
Examples Find f(g(x)) and g(f(x)) and their
domains.
10,12,14
9
Example Find f(g(x)) and g(f(x)) and their
domains.
10
Decomposing Functions

• Examples 16-20 evens

11
Relation

• A set of ordered pairs.

12
Is an ordered pair in the relation defined by

• Plug the x-value in for x and the y-value in for
  y and solve to determine if the statement is true.

13
Example

• Pg. 128 26

14
Implicitly Defined Functions

• Functions that cannot be clearly written as y.
  They are more easily written in the form
  variables constant.
• Example x2 y2 5

15
Example

• Pg. 128 28,30

16
Defining Functions Parametrically

• A parametera variable used to create rules to
  find x and y.
• The variable will typically represent time.

17
Putting the Calculator in Parametric Mode

• Press the mode button.
• Arrow down to Func Par Pol Seq.
• Place the cursor on Par(Parametric) and hit
  enter.
• Go to Y.

18
Example

• Pg. 128 32-38 evens

19
Definition of Inverse Relation

• The ordered pair (a,b) is in a relation if and
  only if the ordered pair (b,a) is in the inverse
  relation.

20
Relation or Function?

• If a graph passes the vertical line test, then it
  is a function.

21
Is the inverse relation a function?

• The inverse of a relation is a function if and
  only if each horizontal line intersects the graph
  of the original relation in at most one point.

22
Horizontal Line Test

• The inverse of a relation is a function if and
  only if each horizontal line intersects the graph
  of the original relation in at most one point.

23
A function Whose Inverse is a Function

• Its graph will pass both the vertical and
  horizontal line test.

24
One-to-One Function

• Every x is paired with a unique y and every y is
  paired with a unique x.

25
Inverse Function Definition

• If f is a one-to-one function with domain D and
  range R, then the inverse function of f, denoted
  f-1, is the function with domain R and range D
  defined by
• f-1(b) a if and only if f(a) b

26
Inverse Reflection Principle

• The points (a,b) and (b,a) in the coordinate
  plane are symmetric with respect to the line yx.
  The points (a,b) and (b,a) are reflections of
  each other across the line yx.

27
The Graph of f-1

• The graph of f is reflected over the line yx to
  get the graph of f-1.

28
Examples Is the relation a function? Does it
have an inverse?

• 40,42

29
The Inverse Composition Rule

• A function f is one-to-one with inverse function
  g if and only if
• f(g(x)) x for every x in the domain of g, and
• g(f(x)) x for every x in the domain of f.

30
Finding the Inverse Function Algebraically

• Determine that there is an inverse by ensuring
  the f is one-to-one. If needed, state any
  restrictions on the domain to make it one-to-one.
• Switch xs to y and ys to x.
• Solve for y to get the formula for the inverse.
  State any restrictions on the domain of the
  inverse.

31
Examples Finding an Inverse

• 44-62 evens