How to find a function of f with the following characteristics

[dazza555]

Homework Statement

Give an expression for a function f which has all the following characteristics:

f (0) = 4 , and
f is increasing for x < 0 , and decreasing for x > 0 , and
y = 1 is a horizontal asymptote of the graph of f as x→-∞ , and
y = 0 is a horizontal asymptote of the graph of f as x→∞ .

Homework Equations

Any I guess

The Attempt at a Solution

Ok, since part a was to do with trigonometry I figured this being part b might follow suit but after having tried 4cos(x), 4/cos(x) and several other differing equations revolving around some manipulation of cos(0)=1 I've come up totally empty.

So then I tried all manner of sorts of x+4 and 1/x +4 but still nothing.

I'm not even sure of where to exactly start or how to approach this question aside from trial and error. I can find several equations to fit each characteristic but combining them is stumping me. The only other thing I can think of is a piece wise equation but again I'm not sure how to approach it or where to start.

Tips and hints on where to begin are all I'm after, as long as I know I'm on the right path.

 

[tiny-tim]

hi dazza555!

The only other thing I can think of is a piece wise equation but again I'm not sure how to approach it or where to start.

piecewise is fine

(nothing in the question requires the answer to have a continuous derivative)

and can you think of a function with f(0) = 0 and f(+∞) = 1 ?

 

[dazza555]

Thank you tiny-tim, that helps quite a bit. I shall work on it some more and post back early tomoz of how I go.

 

[dazza555]

Ok, so I've come up with the following piecewise equation:

f(x)= 4 / e^(x^2), x>=0

1 - 1/e^(x^2), x<0

I'm fairly confident it satisfies all the criteria but I know I'll feel better once somebody confirms it (especially the "f(x) is increasing for x < 0 , and decreasing for x > 0" part).

 

[tiny-tim]

hi dazza555!

(try using the X² icon just above the Reply box )

f(x)= 4 / e^x2, x>=0

1 - 1/eSUP]x²[/SUP], x<0

(wouldn't it be better to write them as 4e^-x2 and 1 - e^-x2 ?

and why not use x or -x instead of x² ?)

the first one is ok, but the second is decreasing, from 1 at -∞ to 0 at 0

(and they aren't continuous at 0 … i know the question doesn't actually specify that, but it would be neater if it was continuous)

 

[dazza555]

I'm an idiot, I can't believe I missed that bit on the second piece, I've just assumed graph go up, graph increasing. Well it's a simple fix, if I change my second part to 1+3e^-x2 it's all fixed and it'll be continuous at (0,0).

The reason I chose x² is because it gives a nice bell shape curve rather than a sharp spike up. Since I'm graphing it also on a rather small axis I figured this way I won't get marked down because it looks linear. Also the reason I wrote it as 4/e^(x^2) was because I failed to notice the superscript button and it kinda looked confusing with little hats going everywhere. But the constructive criticism is welcome, without it I never would've looked for it

 

[dazza555]

Also thanks for all the help tiny-tim.