What Are the Solutions to These Function Analysis Problems?

[ScienceNewb]

Studying for exams and these few questions are stopping me from proceeding, got to learn it all by... (looks at date)... tomorrow or I'm basically screwed.

1. State domain and range of f(x) 2-|x-1|

2. Let g;[b,2] -> R where g(x) = 1 - x^2 , if b is the smallest real value of such that g has an inverse function, find b and inverse of g(x)

3. let f:S->R. f(x) = square root (4-x^20) and S be the set of all real values for x for which f(x) is defined. Let g:R->R, where g(x) = x^2 + 1

Find S and range of f and g.

4. Let 'a' be a positive number, let f:[2, infinite) -> R, f(x) = a - x and let g:(-infinite, 1] ->R, g(x) = x^2 + a. Find all values of 'a' for which f(g(x)) and g(f(x)) are defined.

5.Sketch this hybrid function (just want to know how to go about doing this)
f(x)=
{2x+6 where 0<2x≤2
{-x + 5 where =4≤x≤0
{=4 where x < -4

 

[Cyosis]

If you want assistance you will have to show us what you've done so far.

 

[ScienceNewb]

If you want assistance you will have to show us what you've done so far.

Nothing too coherent. These are the questions I have left after doing just about everything else

 

[Cyosis]

Then start with question one and explain where you get stuck.

 

[ScienceNewb]

Then start with question one and explain where you get stuck.

I get stuck with all of these hence why I chose to put them up on this site as a last resort.

For q.1 I don't understand what to do with absolute values

 

[Cyosis]

Do you know what a domain is and how you usually find it?

 

[ScienceNewb]

Do you know what a domain is and how you usually find it?

All possible x values, depends on the equation in terms of how I find it I guess

 

[Cyosis]

Yes that's right. Usually you look at the function and check if there are any x values that cause problems. For example for the function 1/x you know that x cannot be 0 so the domain will be all values except 0. What are the possible values for 2-|x-1|?

 

[Gigasoft]

Well, in my opinion some of these questions are not very well formulated.

For question 1, the domain and image cannot be written out in a meaningful way unless one additionally restricts the set of possible values for x to R, C or some other set, and also specifies the codomain for f.

Question 3 should probably be: let f:S -> R. f(x) = h(x), with h(x) = sqrt(4-x^20) for all x in R if defined, and with codomain R, and S bet set of all real values for x for which h(x) is defined etc. Otherwise, S is defined in terms of itself.

To answer question 4, express the ranges of f and g in terms of a, exclude the domains of g and f, and find the set of values for a for which this becomes the empty set, by considering the value for a at which point the set changes between empty and nonempty.

To answer question 5, determine each subdomain of f, and sketch each part separately for values of x within that subdomain.

Finding the range of a function involves finding the maximum and minimum of the function for each interval in which it is continuous, taking account the endpoints of the input interval and noting whether the range interval is open or closed. Then, one unions these intervals.