Jan 14 - Jan 18 Flashcards
New Functions from Old Functions Expontential Functions Inverse Functions Logarithims
Vertical and Horizontal Shifts
y = f(x) + c = shifts graph up y = f(x) - c = shifts graph down y = f(x - c) = Shifts graph right y = f(x + c) = Shifts graph left
Given two functions f and g, the composite function f o g is defined by
( f o g) ( x) = f(g(x))
An exponential Function
f (x) = b ^ x
Laws of Exponents
If a and b are positive numbers and x and y are any real numbers, then
- b^ x+y = b^ x b^ y
- b^ x-y = b^ x/ b^ y
- (b^ x)^ y = b^ xy
- (ab)^ x = a^ x b^ x
Natural Exponential Function
f(x) = e^ x
e = 2.71828
When is a function 1 to 1
If it never takes on the same value twice, that is
f(x1) cant = f(x2)
so f(x) is 1 to 1 g(x) is not
Horizontal Line Test
A function is one-to-one if and only if no horizontal line intersects its graph more than once.
Definiton of Inverse Functions (one-to-one)
Let f be a one-to-one function with domain A and range B, Then its inverse function f^ -1 has domain B and range A and is defined by
f^ -1 (y) = x is the same as f(x) = y
Cancellation Equations
f^ -1(f(x)) = x for every x in A
f(f^ -1(x)) = x for every x in B
How to Find the Inverse Function of a One-to-One Function f
Write y = f(x)
Solve for x in terms of y.
Express f^-1 as a function of x, interchange x and y.
Results in y = f^ -1 (x)
Inverse Function of Logarithmic Functions
logb x = y is the same as b^ y = x
Cancellation Equations for Logarithmic Equations
logb(b^ x) = x for ever X E R
b^ logb x= x for every x > 0
Laws Of Logarithms
If x and y are positive then
- logb (xy) = logb x + logb y
- logb (x/y) = logb x - logb y
- logb(x^ r) = rlogb x
Natural Logarithm Notation
logb x = ln x
if we put b = e and replace loge with ln it becomes
ln x = y i the same as e^ y = x
ln(e^ x) = x X E R
e^ lnx = x x > 0
Changes of BAse Formula
For any positive number b (b DOES NOT = 1) we have
logb x = ln x/ ln b
