Exploring Function Bijections: An Example and Proof from [0,1] to [a,b]

[sandra1]

Homework Statement

I really don't understand the question for this problem, could you please help me out? Thanks so much

1.a,b are some real numbers. Give an example of a bijection from [0,1] to [a,b]
2. Prove that all functions from [0,1] to [a,b] are bijections

Homework Equations

The Attempt at a Solution

I think the example function could be something like Sqrt(1-x) + Sqrt(x). That makes its domain be [0,1]. But I don't know how to make it end up with the value of [a,b]

 

[Mark44]

Try a function like this--f(x) = c(x - h)--with domain restricted to [0, 1]. You'll need to figure out what the constants c and h need to be so that f(0) = a and f(1) = b.

 

[sandra1]

Hi, oh yes, your function works. All functions f(x) = mx + n are bijections. Thanks for your help.

About the second one, do you think I should break it into 3 cases. With a,b < 0; a,b > 0; and a <0 ^ b>0?

 

[Mark44]

Before looking at your cases, you should first ask yourself whether it's true. "All functions" covers a lot of territory, including functions that are continuous as well as those that are discontinuous.