Section+1.5

1.5 **//__Inverse Functions__//** Review: Inverse Operations- Inverse Function: 8-3=5 f(x)= 3x-1 5+3=8 f(5)= 14 f(14)=5 (inverse) The ordered Pair (a,b) is in a relation if the ordered pair (a,b) is in the inverse relation. Ex. y=x^2 (-2,4) (-1,1) (0,0) (1,1) (2,4) (function) Inverse if y=x^2 (4,-2) 1,-1) (0,0) (1,1) (4,2) ( not a function, but an Inverse Relation)
 * Inverse Relation:**

A __Function__ where every y-cooridinate maps to a unique x- cooridinate. If a __Function__ is __one-to-one__ than it has an inverse __function__. If a horizontal line only intersects the graph in at most **one** point, than it passes this test. If an equation passes **both** the vertical and the horizontal line test, then it is __one-to-one__. Ex. Relation, Function, or One-to-One? [|Relations.bmp] For Domain and Range, When Looking for the inverse Domain and Range, The Range becomes the Domain and the Domain becomes the Range. Sunch as f(x) =3x+5= D: -2 R:-1 The inverse: f^-1(x)=x-5/3 R:-2 D:-1 1.) Determine that the function is one-to-one, and necessary, state any domain restrictions. 2.)Switch x and y.3.) solve for y, and state any restrictions on the domain of f^-1(x). Ex. F(x)=3x+5 **Ex. f(x)=x/x+1** x=3y+5 //**x=y/y+1**// x-5=3y **x(y+1)=y** x-5/3=y **xy+x=y** f^-1(x)= x-5/3 **x=y-xy x=y(1-x) (distribute) //f^-1(x) = x/1-x//**
 * One-to-One**
 * Horizontal Line Test:**
 * Inverse Function:**
 * Finding An inverse Function:**

A Relation and its inverse are symmetric over the y=x line. [|relation graph.bmp]
 * Graphically**:

Show that f(g(x)) + g(f(x)) =x Ex. Verify that f(x)=x^3+1 and g(x) =cube root of x-1 are inverses F(g(x)) **g(f(x))** f(cuberoot of x-1) G(x^3+1) (cube root of x-1) ^3+1 **(cube root of x^3+1) -1** x-1+1=x **cube root of x^3 =x**
 * Verifying Inverses**:


 * //__Homework: Pg 135 #13-31 (odd)__//**