Fractions #27 - Questions and Answers

Question 1 I have been quite confused by the definition of functions and their uses. First of all can one define functions in a clear understandable way, with a clear explanation of their uses, how they work and what they do? Also I have some specific questions regarding functions Let me give you a example: y = f(x) => This is one of the main reasons I have difficulties understanding functions.... what does the above statement tell me, and if y is a function why do we use y = at all for a formula like y = mx + b would it be the same as writing f(x) = mx + b? Something like y = x^2 is apparently a function... but where is the function name? Which is the input and which is the output? Lastly another example: Let me suppose f = distance f(t) = t^2 f(2) = 4 => Does this mean distance is 4... which is the input which is the output? Mainly what I’m looking for in an answer is a clear and ‘easy’ explanation with examples to help me understand this new topic Answer 1 I see there are already a few answers to this question but I’m going to try my hand at answering it the way I see it. A function f is a mathematical object that relates elements of two sets, one called the domain A and one called the codomain B. The notation f : A → B denotes the fact that f is a function with domain A and codomain B. What it means to have a function f : A → B is this: f assigns to each element of A exactly one element of B. If a ∈ A, the notation f(a) denotes the element of B to which a is assigned by f. Those elements of B which can be written in the form f(a), for some a ∈ A, are called values of f. The set of all values of f is the image of f, which is a subset of B (and not necessarily equal to B). Here are various ways of specifying functions. For example: If A is finite, you can simply list the values of f. For example we can define a function f : {1, 2, 3} → {red, green, blue} by f(1) = green, f(2) = blue, f(3) = red Non-numeric examples of functions are often defined by an equation. An example of a function f : R → R is the one defined by the equation f(x) = x^2 + 3 This equation f is a function. What it means is, given an element x ∈ R, the value of R associated by f with x is x^2 + 3. The expressions f(x) and x^2 + 3 both denote exactly the same thing, namely the value of f associated with the number x. For example f(2) denotes the same thing as 2^2 + 3 which in turn denotes the same thing as 7. The function is f, and f(x) denotes the value of f at a given number x. The graph of a function f : A → B is the set {(a, b) ∈ A x B : b = f(a)} Sometimes, particularly when f : R → R, it is convenient to define a function in terms of its graph. (Indeed, R x R what you’re depicting when you draw a pair of coordinate axes.) For example the equation y = x^2 + 3 specifies a function f : R → R whose graph is the set {(x, y) ∈ R x R : y = x^2 + 3} That is, the function f specified by this equation is the one which associates to each x ∈ R the value x^2 + 3. Some people would then say y is a function of x, but this is slightly misleading: what it means is that there is a function whose values are exactly the values of y satisfying this equation. An example of how a function works is as follows. Suppose a bird is flying in a straight line at a constant speed of 12 metres per second. The distance behind the bird flies is a function of the time it has been since it is. If t is a positive real number, then the distance flown by the bird in t seconds is 12t metres. Thus the relationship between distance and time defines a function f : R+ → R, which is defined by the equation d(t) = 12t for all t ∈ R+. Thus you can consider d as being the ‘distance function’, and for each t you can consider d(t) as the ‘distance travelled at time t’. Some non-examples of functions are: f : R → Q defined by f(x) = ±√x. This is not a function because, for example, f(4) = ±2, which is not an element of Q. f : R+ → R defined by the graph x^2 + y^2 = 1. This is not a function because the values of y are not uniquely determined by the values of x, for example 0^2 + 1^2 = 1 and 0^2 + (-1)^2 = 1. In summary, if f : A → B is a function, then f is the function itself, with its domain A and codomain B; f(a) = (expression in terms of a) is an equation which specifying f by declaring its effect on the elements of A; the expression f(a) is not itself the function f (f is the function), but a function is determined by what its values, so specifying f(a) suffices; y = f(x) is an equation that specifies f in terms of its graph. Answer 2 A function requires some inputs and for each valid combination of inputs produces one output. The output might be determined by the domain, which is sometimes specified but sometimes left ambiguous. Here, the issue is when talking about graphs, because historically graphs of functions use letters to refer to changing quantities, and still do so in many areas of mathematics. When we say that ‘y is a function of x’ it means that Y changes together with X. That concept does not correspond directly to the proper definition of functions. As you realize, when you have a straight line it can be described by the linear function f where f(x) = mx + c f or any x ∈ R, with m,c as some fixed values. Notice that everything here is fixed. f is a fixed unchanging object called a function. ‘f(x)’ is a notation to denote the output when x is given as input to f. To use your other example, if f is a function such that f(t) = t^2 for any t ∈ R, then f(0) = 0^2 = 0, and f(2) = 2^2 = 4 and so on. This shows that if you’ve a different input to a function, you may get a different output. But the function itself hasn’t changed. It always does the same thing. If you input the same input many times in a row, it is going to give you the same output each time. Hence the input is changing. You can give it different numbers, but it is not f itself that ‘changes’. Note also that the variable used as the input in defining a function is not important and is called a dummy variable. In the above example you could have defined f to be such that f(u) = u^2 and it would make no difference. So y = f(x) and y = x^2 are actually misleading notations because it is used to denote y changing when x is changed. I wouldn’t use it as far as possible. If you really need to write something like ‘y = x^2’, you should understand it as follows: ‘When x = 3, y = 9. When x = -2, y = 4’. Note that such notation is often used in the context of plotting graphs, but then remember that these are just equations describing what points are in the graph, and equations are not the same as functions. Answer 3 I’m going to take a little bit of a different approach from that commonly found in textbooks, and instead of giving you mathematical explanations and examples, I’ll try to explain it in terms of the semantics and usage of functions. It seems to me that this where the confusion actually lies (not just for you – for most other people too.) First off, the definitions others have posted here and that you probably can find in your textbook are correct. I’ll put it into layman’s terms: That’s a pretty broad definition, which probably doesn’t sound too useful, and which also probably doesn’t go far in explaining all of the y = x^2 and f(x) = x^2 stuff you are seeing. So let’s think about this definition a little bit first. What we are talking about is an abstract idea – we are talking about the concept of a ‘thing’ which takes in information and gives back some other information. What makes a function special is the requirement that it has to be predictable – one input can’t give you more different outputs. Question 2 How do you convert 5% into a fraction and decimal? Answer 1 5% as a fraction is: 5/100 or 1/20 5% as a decimal is: 0.05. Answer 2 5/100 or 1/20 is a fraction, and 0.05 is a decimal. Answer 3 ‘Percent’ or ‘%’ means ‘out of 100’ or ‘per 100’. It means that 5% can be written as 5/100 = 1/20 = 0.05. Question 3 How do you change 1.25 into a fraction? Answer 1 5/4 Explanation: To convert decimals to fractions I like to turn it into a percent first. 1.25 in percent is 125%. 125% is 125/100 in fraction. Reduce that by 25 and you get 5/4. Answer 2 As 1.25 has 2 numbers after the decimal point, we will divide it by 100 in order to remove the decimal. Hence, we get: 125/100 Now, dividing both sides by a common factor of 25, we will get: 125/100 = 5/4 Thus, 1.25 as a fraction is 5/4. Question 4 How do you convert 1/8 into a decimal? Answer 1 Explanation: To convert a fraction to a decimal, you must do the division indicated by the fraction bar. 1 ÷ 8 = 0.125 Answer 2 To convert 1/8 to a decimal, divide the denominator into the numerator. 1 divided by 8 = .125 Answer 3 We just count how much 1 will be divided by 8, we get the answer 0.125. Question 5 What is the fraction 1/5 as a percent? Answer 1 To the fraction 1/5 as a percent is 20%. Explanation: