lecture01 - 1 1.1 Functions What is a function All a...

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1 Functions 1.1 What is a function? All a function is, is something that takes a number and turns it into another number. Example 1.1. Remember from geometry class the formula for a circle, A = πr 2 . This is a nice example of a function. It takes a number, r , and spits out another number, A . Example 1.2. For a function, we don’t need a simple formula. The cost, C , of mailing a letter that weights w follows a very specific formula, but it’s not as nice as the formula for the area of a circle. Example 1.3. Functions don’t even need to have formulas. Think of the pop- ulation of the world for a given year. This takes a number, the year, and spits out another number, the population. There’s no real formula, however. And we can’t plug in future years and expect to get back an answer that’s even close Now for a better definition Definition 1. A function f is a rule that assigns to each element x in a set A exactly one element, called f ( x ), in a set B . What does this mean? Look at this diagram: The set A is called the domain of f and represents every value you can plug into f and get out another number. Now, a function may not hit every number in the set B . But the set of points that f ( x ) takes on is called the range of f . A symbol that represents an element of the domain is called the independent variable. A symbol that represents an element in the domain is called a dependent variable. Example 1.4. Let’s go back to the area of a circle, A = πr 2 . Sure, we can plug any number into r , but remember that it represents a length, so we’re not going to be plugging in any negative numbers. Anything else goes, however, so the domain is all positive real numbers. Also, A is an area, so it will certainly 1
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never be negative and thus the range will be all positive numbers. Based off this formula, the area depends on the radius, so r is the independent variable and A is the dependent variable. Another way to find the domain is to think of the points you can’t plug in. Try to find the points where, if you were to plug them in, the function “breaks”. This is usually most obvious when the function is represented as a formula. Example 1.5. a) f ( x ) = x + 5, b) g ( x ) = 1 x 2 - x I say it’s more obvious because you can look at the function and you can usually tell the basic idea of where it will break. You can use this to find nice formulas where it breaks. So what about those two? a) We know that we can only take the square root of a positive number. So whatever is under the square root symbol has to be bigger than or equal to 0. We write this as x + 5 0. We can solve this inequality to get x ≥ - 5. We write this as either x ≥ - 5 or using the interval notation [ - 5 , ) b) Well, with fractions, we know that we can’t divide by zero but anything else is just fine. So we’re only worried about where the denominator is 0. We write this as x 2 - x = 0, or x ( x - 1) = 0. This is easy to solve and find that x
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lecture01 - 1 1.1 Functions What is a function All a...

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