lecture03 - Lecture 3 Functions and Limits 3.1 Functions In...

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Lecture 3 - Functions and Limits 3.1 Functions In the expression y = f ( x ) = x 2 + 3 we say that x is the independent variable and that y is the dependent variable . A function is a relationship between two variables such that to each value of the inde- pendent variable there corresponds exactly one value of the dependent variable. This is a rewording of what you may know as the so-called “vertical line test” for functions. The domain of a function is the set of all values of the independent variable for which the function is defined. Examples: 1. Find the domain of f ( x ) = x 2 1 - x solution: In a question like this, we are trying to find the largest possible domain. Here, the only potential problem is that the denominator could be 0. So the domain is every real number except 1. We write Domain = { x | x 6 = 1 } We will see later that the graph looks like this:
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2. Find the domain of f ( x ) = 2 x - 3 solution: Here we need to make sure that what’s inside the square root is non-negative. So, 2 x - 3 0 x 3 2 So the domain is 3 2 , ) . Note that 3 2 is included. The graph looks like: Related to the domain is the range , which is the set of all y -values a function could possibly take. Composites If f and g are functions, then their composite, denoted f g , is defined to be f ( g ( x )). In other words, apply g to x and then apply f to the result. Example: If f ( x ) = 1 + x 2 and g ( x ) = 2 x - 1, then f ( g ( x )) = f (2 x - 1) = 1 + (2 x - 1) 2 = 1 + (4 x 2 - 4 x + 1) = 4 x 2 - 4 x + 2 On the other hand, g ( f ( x )) = g (1 + x 2 ) = 2(1 + x 2 ) - 1 = 2 x 2 + 1
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So f g 6 = g f , in general. Inverses Two functions f and g are inverses of each other if f ( g ( x )) = x and g ( f ( x ) = x . Normally, g is denoted f - 1 . So, f f - 1 ( x ) = x f - 1 f ( x ) = x Example: If f ( x ) = 6 - 3 x , then I claim that f - 1 ( x ) = - 1 3 x + 2.
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