Math_137_Winter_2010_Week_2_Notes

Math_137_Winter_2010_Week_2_Notes - Math 137 Week 2 Notes...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 137 Week 2 Notes – Dr. Paula Smith I. Exponential Function A. Properties of the function y = a x , where a > 0. (No exponential is defined for a < 0.) 1. Domain = , Range = {x | x > 0} 2. Graph, for a > 1. If 0 < a < 1, then there is a b = 1/a > 1, and f(x) = a x = (1/b) x = b -x , and the graph will look like the reflection of the above graph across the y-axis. 3. a 0 = 1 for any a > 0. 4. a x a y = a x+y and a x / a y = a x – y 5. (a x ) y = a xy 6. a x b x = (ab) x 7. The exponential function with base e = 2.71828182845… is the one for which the slope of its graph at x = 0 is exactly 1. II. Inverses of Functions A. Composition 1. The operation f(g(x)) = f ° g (x) is called composition. Dom f ° g is the set of all x in Dom g such that g(x) is in Dom f. 2. In general, f ° g (x) g ° f (x), but for some pairs of functions f ° g (x) = x for every x in Dom g and g ° f (y) = y for every y in Dom f. B. One-to-one functions 1. A function f is one-to-one if f(x) f(z) whenever x z. 2. The Horizontal Line Test states that a function is 1-1 iff no horizontal line intersects its graph more than once. 3. If a function f is 1-1, then there is a function g = f – 1 such that f ° f – 1 (y) = y for every y in Dom f – 1 and f – 1 ° f (x) = x for every x in Dom f. f – 1 is called the inverse (under composition) of f. 4. 1/f is the inverse of f under multiplication. 1/f(x) is not f – 1 (x), with the exception of f(x) = 1/x. C. Inverses
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
1. If f(a) = b, then f – 1 (b) = a. Thus if (a,b) is on Graph f, then (b,a) is on Graph f – 1 . Hence the graph of an invertible function is the reflection of the original function across the y = x line. 2. If f is not 1-1, in many cases we can restrict the domain of f to create a function that is 1-1 x 2n is not 1–1, but x 2n restricted to x 0 is 1–1. For this restricted function, Domain = {x | x 0} = Range. As the inverse of this restricted function, the graph shown is the re ection of the right half of y = x 2n across the line y = x. 3. To find the inverse function of a 1-1 function: a) Write y = f(x). b) Isolate x in terms of y. c) Interchange x and y; the result is y = f – 1 (x).
Background image of page 2
4. Dom f = Ran f – 1 and Dom f – 1 = Ran f. III. Logarithms A. Log a x as inverse of a x 1. The exponential function f(x) = a x is 1-1, and therefore has an inverse, f – 1 (x) = log a x. y = log a x means x = a y . 2.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 9

Math_137_Winter_2010_Week_2_Notes - Math 137 Week 2 Notes...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online