What Are the Domain and Range of Transformed Functions?

[coco richie]

Homework Statement

In exercises 39-46, f refers to the function with domain [0,2] and range [0,1], whose graph is shown in Figure P.55 (how do I draw and paste graphs? the graph looks like an upside down curve with a maximum height or range of (1,1) and a width or domain of (0,2)). Sketch the graph of the indicated functions and specify their domain and ranges.

Homework Equations

39. f(x) + 2
41. f(x + 2)
43. -f(x)
45. f(4 - x)

The Attempt at a Solution

No idea.

 

[eumyang]

Look in your book for information on graphical transformations of functions (or something like that) and READ the material. Then try these problems again and show us your attempt of the solution.

 

[coco richie]

Is there anyone who can help me with a couple of these problems so that I can get the gist of it? How do I interpret something like f(x) + 2? I've never came across such a problem in my life? Please help.

 

[eumyang]

If "helping you" means give you answers, we can't, because that would be against the forum rules. However, I will get you started, but in doing so I may be violating the forum rules.

Suppose f(x) is another function, $$f(x) = \sqrt{x}$$. I assume you know how to graph this.) The domain would be [0, ∞) and the range would be [0, ∞). If we look at f(x) + 5, this would mean $$f(x) + 5 = \sqrt{x} + 5$$, so if you graph this, you can see that the new graph has shifted up 5 units. This does not change the domain, but the range will change. The domain of f(x) + 5 is still [0, ∞), but this time the range is [5, ∞). Now look at f(x - 8), or $$f(x - 8) = \sqrt{x - 8}$$. If you graph this, the graph will shift 8 units to the right of f(x). The range will be the same, but this time the domain will be different. The domain of f(x) is [0, ∞), but the domain of f(x - 8) is [8, ∞). The range of both functions will be the same, [0, ∞). Again, you really should read the section in your book about graphical transformations. They'll probably use a letter as part of their notation, like f(x) + c ("c" is what my book uses). Or heck, you could probably Google all of this.

 

[HallsofIvy]

Anything of the form af(x)+ b- that is where all changes are outside the function- changes only the range.

Anythihng of the form f(ax+ b)- that is where all changes are inside the function- changes only the domain.