Chapter 1: Functions and Graphs

1. Chapter 1:Functions and Graphs Section 1-5: Parametric Relations and Inverses

2. Objectives • You will learn about: • Defined relations parametrically • Inverse relations and inverse functions • Why: • Some functions and graphs can best be defined parametrically, while some others can be best understood as inverses of functions we already know.

3. Vocabulary • Parametric • Inverse relation • Inverse function of f • Reflections

4. Relations Defined Parametrically • Another way to define functions or relations is to define both elements of the ordered pair (x, y) in terms f another variable t, called a parameter.

5. Example 1Defining Relations Parametrically • Consider the set of ordered pairs (x, y) defined by the equations: • x = t + 1 • y = t2 + 2t • Where t is a real number. • Find the points determined by -3, -2, -1, 0, 1, 2, 3. • Find an algebraic relationship between x and y. • Graph the relation in the xy plane

6. Example 2Using a Graphing Calculator in Parametric Mode • Consider the set of all ordered pairs (x, y) defined by the equations: • x = t2 + 2t • y = t + 1 • Use a graphing calculator to find the points determined by t = -3, -2, -1, 0, 1, 2, 3. • Graph the relation in the xy plane. • Is y a function of x? • Find an algebraic relationship between x and y.

7. 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.

8. 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.

9. Example 3Applying the Horizontal Line Test • Which of the graphs are graphs of: • Relations that are functions. • Hint: Use the vertical line test. • Relations that have inverses that are functions. • Hint: Use the horizontal line test.

10. Inverse Functions • 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. Example 4Finding an Inverse Function Algebraically • Find an equation for f -1(x) if:

12. The 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 (a, b) and (b, a) are reflections of each other across the line y=x.

13. Example 5Finding an Inverse Function Graphically • The graph of a function y=f(x) is shown. Sketch a graph of the function y=f -1(x). Is f a one-to-one function?

14. Inverse Composition Rule • A function is one-to-one with inverse function f 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

15. Example 6Verifying Inverse Functions • Show algebraically that the two functions are inverses:

16. How to Find an Inverse Function Algebraically • Given a formula for a function f, proceed as follows to find a formula for f -1: • Determine that there is a function f -1 by checking that f is one-to-one. State any restrictions on the domain of f. • Switch x and y in the formula y = f(x). • Solve for y to get the formula y = f -1(x). State any restrictions on the domain of f -1.

17. Example 7Finding an Inverse Function • Show that the function f has an inverse function and find a rule for f -1(x). Show any restrictions on the domain of f and f -1.