Solving Functions f and g: Range and Relationship with Homework Statement

[WeiLoong]

Homework Statement

The functions f and g are given by
https://scontent-kul1-1.xx.fbcdn.net/hphotos-xtf1/v/t35.0-12/11993948_10204837479208369_1887096410_o.jpg?oh=b1653a61128c571af8137b1fd00ccb01&oe=55F47BC4
a) without using differentiation, find the range of f
b)show that f(x)^2+g(x)^2=1.Hence find the range of g.

Homework Equations

FUnction

The Attempt at a Solution

[/B]
for question A i used limit to infinity so that the function can be a number, thus I got x=all real number if i use limit to infinity. for question B i have no clue. please help

 

[RUber]

I agree. f and g are both defined for all real values of x.
For problem b, what do you get when you square these functions?
what is [f(x)]^2, what is [g(x)]^2?
Note that they both have the same denominator, so you need to show that the sum of the squared numerators is equal to the square of the common denominator.

Do you know what ##(e^x)^2 ## is?
Do you know what ##( a + b) ^2 ## is ?

If you know those rules, this just needs to be expanded out to show the answer.

 

[Mark44]

Homework Statement

The functions f and g are given by
https://scontent-kul1-1.xx.fbcdn.net/hphotos-xtf1/v/t35.0-12/11993948_10204837479208369_1887096410_o.jpg?oh=b1653a61128c571af8137b1fd00ccb01&oe=55F47BC4
a) without using differentiation, find the range of f
b)show that f(x)^2+g(x)^2=1.Hence find the range of g.

Homework Equations

FUnction

The Attempt at a Solution

[/B]
for question A i used limit to infinity so that the function can be a number, thus I got x=all real number

Part a asks for the range -- the set of function values -- not the domain.
For ##\lim_{x \to \infty}f(x)##, what did you get? Also, you might want to look at the limit as x approaches -∞. These limits might give you an idea about the behavior of f.

if i use limit to infinity. for question B i have no clue. please help

Did you try part b? It asks you to show that ##(f(x))^2 + (g(x))^2 = 1##. They have told you what the answer should be.

 

[Mark44]

I agree. f and g are both defined for all real values of x.

This isn't relevant, though, as the problem asks for the range of f, not the domain.

 

[HallsofIvy]

Since [itex]e^x[/itex] is positive for all x, [itex]\frac{e^x- e^{-x}}{e^x+ e^{-x}}[/itex] is a fraction with the numerator smaller than the denominator.