Thread: functions and limits

I have trouble with this questions

f:[3,infinity)-->R f(x)=x^2 -6x +10

The answer for this (in mcqs) is
f^-1:[1,infi)-->[3,infi), f^-1(y)= 3+ (y-1)^-.5

I do know how to get the f inverse but i do not know how to get the domains and ranges.
Can you explain what is meant from [1,infi)-->[3,infi) ?

2nd question

f(x)={x x is rational
-x x is irrational

g(x)={sinx x is rational
x^3 x is irrational

Can you comment on the existance of limx->0f(x) and lim->0g(x)

Thank you

PS. sorry about the written format. I don't know how to use maths sybols(is there a link which tells how)

Originally Posted by Dili

2nd question

f(x)={x x is rational
-x x is irrational

g(x)={sinx x is rational
x^3 x is irrational

Can you comment on the existance of limx->0f(x) and lim->0g(x)

Thank you

PS. sorry about the written format. I don't know how to use maths sybols(is there a link which tells how)

(A)

-x <= f(x) <= x

hence as x -> 0 f(x) is trapped between two continuous function both of
which go to 0, so the limit exists and is 0.

(B)

for |x|<1

-x <= g(x) <= x,

so again the limit of g(x) as x->0 is 0.

RonL

Can anyone post an answer to the first question?

Originally Posted by Dili

f:[3,infinity)-->R f(x)=x^2 -6x +10

The answer for this (in mcqs) is
f^-1:[1,infi)-->[3,infi), f^-1(y)= 3+ (y-1)^-.5

I do know how to get the f inverse but i do not know how to get the domains and ranges.
Can you explain what is meant from [1,infi)-->[3,infi) ?

You have a function f with a domain and a given range which is all real numbers.

I don't know what "mcqs" is supposed to be, but it looks like you are trying to find the inverse function.

I'm going to first change the form of f a bit. This is not strictly necessary, but will simplify finding the inverse function.








Now to find the inverse function. I'm going to let y = f(x) represent the function. Then x = f(y) represents the inverse function. So I'm going to solve x = f(y) for y:








So the function is given by . (I changed the variable back to y to match with your problem. It's a "dummy" variable and can be anything we wish.)

How about domain and range? Well, the domain of f is so this is the range of . The allowed domain of is . This domain gives the exact range as we expect (it must give a range at least as large as the expected range, else we need to modify the domain such that it will.) So the function may be expressed as:


-Dan