Math_137_Winter_2010_Week_2_Notes

# Math_137_Winter_2010_Week_2_Notes - Math 137 Week 2 Notes...

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

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

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

View Full Document
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).
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.

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

View Full Document
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
Ask a homework question - tutors are online