# Find functions

• October 31st 2008, 05:33 PM
Find functions
1. Find a continuous function $f:(0,1) \rightarrow \mathbb {R}$ with an image equal to $\mathbb {R}$

I'm thinking may be something like arctanx + 1/2 ?

2. Find a continuous function $f:(0,1) \rightarrow \mathbb {R}$ with an image equal to [0,1]

tanx?

3. Find a continuous function $f: \mathbb {R} \rightarrow \mathbb {R}$ that is strictly increasing and has image equal to (-1,1).

Sinx might work but it ain't strictly increasing...
• October 31st 2008, 06:21 PM
Jhevon
Quote:

1. Find a continuous function $f:(0,1) \rightarrow \mathbb {R}$ with an image equal to $\mathbb {R}$

I'm thinking may be something like arctanx + 1/2 ?

do you actually remember what the arctan(x) graph looks like? it is not arctan(x) that should be used here, the range for that graph is certainly not all real numbers, it is $[- \pi / 2, \pi /2]$

you want tan(x) here. but it will be transformed somehow. see if you can figure out how

Quote:

2. Find a continuous function $f:(0,1) \rightarrow \mathbb {R}$ with an image equal to [0,1]

tanx?
here is where you might choose a shifted arctan(x) graph. but consider using something simpler, like a transformed sine or cosine graph

Quote:

3. Find a continuous function $f: \mathbb {R} \rightarrow \mathbb {R}$ that is strictly increasing and has image equal to (-1,1).

Sinx might work but it ain't strictly increasing...
a transformed arctan(x) will work nicely here. sine or cosine won't work