2.6 Relations and Parametric Equations

• Pg. 150 #42 – 44 Pg. 136 #9 – 35 odd

• #25 (3, -4) r = 4 #26 (1, -3) r = 7• #27 (2, -3) r = #28 (7, 4) r =

• #42 [-2, -1)U(-1, ∞) #43 (∞ , -1]U[4, ∞ )• #48 No real solutions #88 y – axis • #89 x – axis #90 no symmetry• #91 origin #92 y – axis, x – axis and origin• #93 origin

2 19

2.6 Relations and Parametric Equations

Circles• Write the standard form of

the equation of a circle from the given information and state the center and radius.

Symmetry Practice• Determine the type of

symmetry for the following equations:

2 2 11 3 13 0x y x y

2 24 12 0x x y y

4 23 5f x x x

5 5 1f x x x

2 3x y x y

23 2 1y x

2.7 Inverse Functions

Inverse Relations• The inverse of relation R,

denoted R-1, consists of all those ordered pairs (b, a) for which (a, b) belong to R.

• In other words, (a, b) is in the relation R if, and only if, (b, a) is in the relation R-1.

• Graphically, an inverse is a reflection of the original graph over the line y = x.

• In generic terms, you find an inverse by swapping the x and y and then solving back for the y.

2.7 Inverse Functions

Examples• Find the inverse of

y = 3x + 8 algebraically.

• Graph the original equation and the inverse along with the line y = x to show it has the proper symmetry.

• Find the inverse of y = x2 algebraically.

• Graph the original equation and the inverse along with the line y = x to show it has the proper symmetry.

2.7 Inverse Functions

Examples• How is f(x) = x2 different

than y = x2?

• In order to find an inverse function of f(x) = x2, we need to set limits on the domain.

Inverse Functions• In order for an inverse

function to exist, first you must be dealing with a function and that function must pass the VLT and the HLT.

• HLT – Horizontal Line Test (of the original function) is just like the VLT, except it will tell you whether or not the inverse will be a function.

2.7 Inverse Functions

Inverse Functions• If an inverse and the original

function are composed together, they should always equal x. This works for all values of x in the domain of each function.– > f-1(f(x)) = x– > f(f-1(x)) = x

Examples• How is f(x) = x2 different

than y = x2?• This fails the HLT, so if you

say f(x) = x2, where x ≥ 0 you are safe! (you can still use y to solve)

• Find the inverse function and prove it is an inverse function.

2.7 Inverse Functions

Inverse Functions• Show that f(x) =

will have an inverse function. – Find the inverse function and

state its domain and range. – Prove that the two are

actually inverses.

• Show that g(x) = will have an inverse function. – Find the inverse function and

state its domain and range. – Prove that the two are

actually inverses.

• Show that h(x) = x3 – 5xwill have an inverse function.

1x 3 4

2

x

x