Chapter 1 Functions & Graphs

Mr. J. Focht

PreCalculus

OHHS

1.5 Parametric Relations & Inverses

Defined Relations Parametrically

Inverse Relations and Inverse Functions

Relations Defined Parametrically

Let’s look at all ordered pairs (x, y ) where x = t + 1 and y = t2 + 2t

Let’s look at all ordered pairs (x, y ) where x = t + 1 and y = t2 + 2t

When x and y are defined by a third variable, they are defined parametrically. t is called a parameter.Parametric relations are most commonly used to track positions (x, y) at certain times t.

Find 6 points in the relation

Let’s look at all ordered pairs (x, y ) where

x = t + 1 and

y = t2 + 2t

t x= t +1 y = t2+2t (x, y)

-3 -2 3 (-2,3)

-2 -1 0 (-1,0)

-1 0 -1 (0,-1)

0 1 0 (1,0)

1 2 3 (2,3)

2 3 8 (3,8)

Find an algebraic relation between

x and yThis is also known as “eliminating the parameter”.

x = t + 1 and y = t2 + 2t

t = x – 1

y = (x-1)2 + 2(x-1)

y= x2 – 2x +1 + 2x – 2

y = x2 -1

Class Work

P. 135, #5

Using a Graphing Calculator in Parametric Mode

x = t + 1 and y = t2 + 2t

Change your calculator from function mode to parametric mode.

Graphing

x = t + 1 and y = t2 + 2t

Enter the two parametric equations

Graphing

Change the range of the graph.

Graph

View the Table

View the table to find a list of (x, y)’s.

Class Work

P. 135, # 7

Inverse Relations

An inverse relation can be found by reversing the order of the pairings.

Vertical Line Test

A relation is a function if it can pass the vertical line test.

Horizontal Line TestThe inverse of a relation is a function if it can pass the horizontal line test.

The inverse of this relation is not a function.

Class Work

P. 135, #9

Vocabulary

If a relation can pass both the vertical and horizontal line tests, both the relation and its inverse are functions.

The relation is called a

one-to-one function.

Inverse Notation

The inverse of a function f(x) is

f-1(x)Be careful. The -1 is not a power. It is part of

the name and does not indicate any action.

It means : if f(a) = b, then f-1(b) = a

Finding the Inverse Algebraically

If f(x) = 2x – 4, find f-1(x).

First rewrite the function as an equation.

y = 2x – 4

Reverse the x and y.

x = 2y – 4

Solve for y

y = ½x + 2 f-1(x) = ½x + 2

Class Work

P. 135, #15

The Inverse Reflection Rule

The graph of the inverse of a function can be found by reversing or inverting the x’s and y’s of the relation.

For every (x,y) in the relation, graph (y,x).

This is a reflection over the line y = x.

Graph the Inverse.

Invert each point.

(-2, 4)

(4,0)

(2,-1)

(-1, 2)

(4, -2)

(0, 4)

(-2, 4)

(-1, 2)

(0, 4) (4, 0)

(2,-1)

(4,-2)

Graph. We don’t need to know the name of the relation.

Class Work

P. 135, #23

The Inverse Composition Rule

g(x) is the inverse function of f(x) if

f(g(x)) = x and g(f(x)) = x

Example

( ) 2 1f x x

Verify that f(x) and g(x) are inverses of each other.

1( )

2

xg x

Testing:

f (g(x))1

22 1x

x

1 1x

g(f(x))

2

2 1 1x

x2

2x

Class Work

P. 136 #27

Homework

P. 136-137

#4, 6, 12, 14, 18, 25, 28, 39-44, 47