September 19, 2014
(1.5) Parametric Equations and Inverses
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
Parametric Equations: equations in which BOTH
elements of the ordered pair (x,y) are defined in terms of
another variable t, called the parameter.
y = -x2 + 5
(path of an object)
tells where an object has been, but not when
We can introduce a third variable , t, called a parameter to
determine WHEN an object was at any given point.
x and y can be written in terms of t.
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
Ex. Consider the set of all ordered pairs defined by the
equations:
x=t
y=t+2
a. Find the points determined by t= -3, -2, -1, 0, 1, 2, 3
b. Graph the relation in the (x,y) plane.
c. Find an algebraic relationship between x and y (eliminate
the parameter).
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
1. Sketch the curve then eliminate the parameter.
t
x = t2 - 4 and y = 2 -2 < t < 3
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
HW:
HR: (1.5) Pg.126: 1-8
September 19, 2014
September 19, 2014
(1.5) Inverse Functions
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
Inverse Relation: The ordered pair (a,b) is in a relation iff the
ordered pair (b,a) is in the inverse relation.
Ex.
Ex. {(1,5),(2,6),(3,7),(4,8)}
Inverse: {(5,1),(6,2),(7,3),(8,4)}
switch the domain and range
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
Inverse Function: If a function is one-to-one, with domain D and
range R, then the inverse function of f, denoted f-1, is the function
with domain R and range D.
Horizontal Line Test:
The inverse of a relation is a function iff each horizontal
line intersects the graph of the original relation in at
most one point.
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
Which of the graphs are graphs of:
a. relations that are functions?
b. relations that have inverses that are functions?
A
B
C
D
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
1. What is the inverse of f(x) = x - 6?
2. What is the inverse of f(x) = 4x?
3. Find the inverse of f(x) = 5x + 1.
Find f(f-1(x)) and f-1(f(x)).
September 19, 2014
Objective: To define relations parametrically.
The Inverse Compostion Rule:
To learn about inverse relations and inverse functions.
A function, f, is one-to-one with inverse
function, g, if and only if:
f(g(x)) = x
and g(f(x)) = x
Verify algebraically that f(x) = 2x3 - 1, and g(x) = x + 1
3
are inverse functions.
2
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
The Inverse Reflection Principle: A function and its
inverse will be reflections over the line y = x.
Sketch the inverse of each function.
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
HW:
HR: (1.5) Pg. 126: 9-33odd, 34-36
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.
1. Find the inverse of
2. Find the inverse of
f(x) = 2x + 5
f(x) =
x+3
x-2
September 19, 2014
Objective: To define relations parametrically.
To learn about inverse relations and inverse functions.