1. Precalculus Chapter 1 Section 5
Parametric Relations and Inverses

2. Relations Defined Parametrically
Another way to define functions, or more generally, relations, is to define both elements of the ordered pair (x, y) in terms of another variable t, called a parameter.
This is useful for science experiments. For example, we could measure an objects flight path by considering its horizontal movement (x) over time (t) and its vertical movement (y) over time (t).
This would yield 2 separate functions related to time (t), for which we could then redefine in terms of just x and y.

3. Defining a Function Parametrically
Consider the set of all ordered pairs (x, y) defined by the equations x = t +1 and y = t2 + 2t, where t is any real number.
A) Find the points determined by t = -3, -2, -1, 0, 1, 2, 3.
B) Find an algebraic relationship between x and y. (This is often called “eliminating the parameter”)
Is y a function of x?
C) Graph the relation in the (x, y) plane.

4. Defining a Function Parametrically
Solution:
A) Substitute each value of t into the formulas for x and y to find the point that it determines parametrically. t
x = t+1
y = t2 +2t
(x, y) -3 -2 3
(-2, 3) -1
(-1, 0)
(0, -1) 1
(1, 0) 2
(2, 3) 8
(3, 8) 4 15
(4, 15)

5. Defining a Function Parametrically
B) We can find the relationship between x and y algebraically by the method of substitution. First solve for t in terms of x to obtain t = x – 1.
y = t2 + 2t
y = (x – 1)2 + 2(x – 1)
y = x2 – 2x x – 2
y = x2 - 1
This is consistent with the ordered pairs we found in the table. As t varies over all real numbers, we will get all the ordered pairs in the relation y = x2 – 1, which does indeed define y as a function of x.

6. Inverse Relations
Definition: the ordered pair (x, y) is in a relation iff the ordered pair (y, x) is in the inverse relation.
Notice how the x & y values switched places?
This is all you really need to know about how to find the inverse relation.
The hard part about this is determining if the inverse relation is a function. We can do this graphically.

7. Inverse Relations
To determine if a relation is a function, we can use the Vertical Line Test of the functions graph.
However, we may not be able to graph the inverse of a function properly.
So how can we determine if the inverse is a function?
If a the function has ordered pairs (x, y) and the inverse has ordered pairs (y, x), how can we determine if the functions inverse is a function?

8. Horizontal Line Test
Definition: The inverse of a relation is a function if and only if each horizontal intersects the graph of the original relation in at most one point.
See graphs of the 12 basic functions to determine if an inverse function exists.

9. One-to-One Functions
Definition: a function is said to be one-to-one if the graph of the function passes both the horizontal and vertical line tests.
That means that every x is paired with a unique y and every y is paired with a unique x.

10. Inverse Functions
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

11. Inverse Reflection Principle
The points (a, b) and (b, a) in the coordinate plane are symmetric with respect to the line y = x.
The points are reflections of each other across the line y = x.

12. Inverse Composition Rule
A function f is one-to-one with inverse function g if and only if
f(g(x)) = x, ∀ x ∈ g; and
g(f(x)) = x, ∀ x ∈ f

13. Homework
Day 1: # 3 – 15 by multiples of 3, and 15 – 20 all on page 135.
Day 2: # 21 –42 by multiples of 3, and 45 – 50 all on pages 135 – 137.