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Confused about: g(x) = (x - 1)^2

  1. Nov 28, 2012 #1
    I am revising on sketching curves. But I am very confused about the following even though I have become very good at sketching graphs.

    An example from the book is:

    Given that:
    i) f(x) = x^3
    ii) g(x) = x(x - 2)

    Sketch the curves with equation y=f(x + 1) and g(x +1)

    I don't understand the example that the book provides. Not because I don't know how to sketch the curve but the use of g(x) is what I don't understand. Somewhere in the example workout it says:

    g(x) = (x - 1)^2
    so -> g(x) = f(x - 2)

    Also
    h(x) = x^2 + 2
    so -> h(x) = f(x) + 2

    Please help as I would like to finish this chapter sooner. The use of h(x) and g(x) is what I don't understand. I will also appreciate any links that you may provide.

    Thanks.
     
  2. jcsd
  3. Nov 28, 2012 #2

    Fredrik

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    I assume that "g(x+1)" really means y=g(x+1). Just use the definition of g (given in ii) to rewrite the right-hand side.

    There's something missing here. This is clearly not the same g as before, so it's probably a different f as well. I can't comment unless you tell me what f is.

    Same thing here. What is f?

    Edit: Some general comments about functions: Students often think of functions as relationships between variables, but it's better to think of a function as a rule that associates a member of a set (called the function's codomain) with each member of a set (called the function's domain). If f is a function, f(x) denotes the member of the codomain that f associates with x. f(x) is called the value of f at x. We also say that f takes x to f(x).

    For example, if f is defined by ##f(x)=x^2## for all real numbers x, then f(3)=9. Here f is the function that takes every number to its square, and its value at 3 is 9. Also, the value of f at x+1 is f(x+1), which by definition of f is =(x+1)2. Note that f(x+1) isn't a function. It's a member of the codomain of f. However, without knowledge of the value of x, we can't know which one it is.

    We can also talk about the function that takes x to f(x+1). If we define g(x)=f(x+1) for all x such that x+1 is in the domain of f, then we have defined a function g. People often refer to that function as "f(x+1)", but if you want to be accurate, you need to refer to it as "the function that takes x to f(x+1)", or something like that.
     
    Last edited: Nov 28, 2012
  4. Nov 28, 2012 #3
    Thanks for the reply. It is beginning to make sense now.
     
  5. Nov 28, 2012 #4
    Hey. This is the first time I've ever helped someone on this site. I'll try my best.
    So is the problem here trying to figure out what these two new f's and g's are supposed to be?

    When you have f(x) = (x)^3, think of the left hand side as saying you are making a rule labelled f on the variable also called your input, or argument, (x). This is denoted f(x). The right hand side says specifically what that rule involving your input is. (x)^3.

    By the way, we could have rules that do a lot more things, like cubing x, then adding another 6x to it, and perhaps adding a constant to it all like 5, then perhaps dividing everything by 2. This would look like f(x) = (x^3 + 6x + 5)/2 . Try to understand this maybe after if it doesn't make sense now.

    Continuing on though...
    y = f(x+1) is saying take your original rule that you were given on (x) ( the rule is to cube (x) right? ), and just do the same rule to a new the input, (x+1), and it also is saying that were calling this new thing y. So in this case we just slap on ^3 to the input (x+1).

    I'm not really sure what you mean by the other functions they are giving you either.

    Hope this helped.
     
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