Invertibility of f(x) and its Derivative g'(y)

[tomboi03]

For x E $$\Re$$, let
f(x) = 1 + $$\int$$et² dt
(the interval for this function goes from (0,x) i just didn't know how to put it on the integral.)

i. Prove that the range of f is $$\Re$$ (i.e. prove that for every y E $$\Re$$ there is an x E $$\Re$$ such that f(x)=y)

ii. Prove that f: $$\Re$$ $$\rightarrow$$$$\Re$$ is invertible

iii. Denote the inverse of f by g. Argue that g is differentiable and show that g satisfies the equation
g'(y) = e^-(g(y))2
for all y E $$\Re$$. Show that g is differentiable twice.

iv. Determine g(1), g'(1), g"(1).

okay, so for i, i have no idea
ii, how can you prove that a function is invertible
iii, i have no idea
iv, i just find the first derivative and the 2nd derivative and find the values of all of that.

Please help me out,
Thank You

 

[tiny-tim]

For x E $$\Re$$, let
f(x) = 1 + $$\int$$et² dt
(the interval for this function goes from (0,x) i just didn't know how to put it on the integral.)

i. Prove that the range of f is $$\Re$$ (i.e. prove that for every y E $$\Re$$ there is an x E $$\Re$$ such that f(x)=y)

Hi tomboi03!

(have an ε )

Do you mean $$f(x)\ =\ 1\ +\ \int_0^xe^{t^2} dt$$ ?

Hint: what are f(-∞) and f(∞)?

 

[tomboi03]

nope, it's still et²

hahahaha :D hehehe
Thanks for helping me! i really appreciate it! :D hehehe

 

[tomboi03]

Can someone help me with this.. the previous person didn't help me much... thank you

 

[Mark44]

For x E $$\Re$$, let
f(x) = 1 + $$\int$$et² dt
(the interval for this function goes from (0,x) i just didn't know how to put it on the integral.)

i. Prove that the range of f is $$\Re$$ (i.e. prove that for every y E $$\Re$$ there is an x E $$\Re$$ such that f(x)=y)

ii. Prove that f: $$\Re$$ $$\rightarrow$$$$\Re$$ is invertible

iii. Denote the inverse of f by g. Argue that g is differentiable and show that g satisfies the equation
g'(y) = e^-(g(y))2
for all y E $$\Re$$. Show that g is differentiable twice.

iv. Determine g(1), g'(1), g"(1).

okay, so for i, i have no idea
ii, how can you prove that a function is invertible
iii, i have no idea
iv, i just find the first derivative and the 2nd derivative and find the values of all of that.

Please help me out,
Thank You

OK, so apparently $$f(x) = \int_0^x et^2 dt$$
Have you gone so far as to evaluate this integral?

 

[tomboi03]

actually it's e^(t^2)...
my professor has changed it.

 

[Tedjn]

Yes, that makes more sense. Which of the questions are you still having problems with?

 

[tiny-tim]

actually it's e^(t^2)...
my professor has changed it.

mmm … that's what i said! …

Do you mean $$f(x)\ =\ 1\ +\ \int_0^xe^{t^2} dt$$ ?

Hint: what are f(-∞) and f(∞)?

ok … for

i. Prove that the range of f is R (i.e. prove that for every y E there is an x E such that f(x)=y)​

… find f(-∞) and f(∞) and describe how it goes from one to the other.

 

[tomboi03]

i'm not sure if i understand how to solve the integral...

and i was wondering... does invertible mean show that it is bijective?

I'm confused.

 

[tiny-tim]

i'm not sure if i understand how to solve the integral...

Hi tomboi03!

You don't need to "solve" it … just look at it … what is its value at ∞?

and i was wondering... does invertible mean show that it is bijective?

Invertible means it has a unique inverse, so if it's onto and invertible, then yes, it's bijective.

 

[tomboi03]

is it infinity? I'm not sure how i can just loook at it... :'(

 

[tiny-tim]

is it infinity? I'm not sure how i can just loook at it... :'(

​

well, if you draw it, the integral is the area under it (to the right of the y-axis) …

isn't that obviously ∞?

('cos it keeps going up! )

ok, now what's f(-∞)? ​

 

[tomboi03]

isn't it negative infiinity?

 

[tomboi03]

so how do you prove that this is invertible?

 

[tiny-tim]

isn't it negative infiinity?

Yup!

so how do you prove that this is invertible?

Well, it goes from -∞ to ∞, so all you have to prove is that it doesn't go through any level twice.