Are these invertible? Why or why not?

  • Thread starter adelaide87
  • Start date
In summary, the functions sech x, cos (ln x), and e^(x^2) are not invertible on the given domains. To prove a function is not invertible, one only needs to find two values of x that result in the same output. On a TI calculator, cos (ln x) can be plotted easily by setting appropriate bounds.
  • #1
adelaide87
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a.) sech x on (0,infinity)

b.) cos (ln x) on (0, e^pi)

c.) e^(x^2) on (-1,2)

I am stuck and have no clue. I have only been able to get through basic questions like this, how do you complete these?

Thanks for any help.
 
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  • #2
whats the definition of invertible? good place to start...

welcome to pf by the way ;)

it generally works here by you having an attempt & and people will help steer you through the problem - though you still do the work
 
  • #3
The usual strategy is to sketch a graph of each function and try to figure out whether each horizontal line intersects the graph at most once. What do you say for those three examples?
 
  • #4
Quick question - how do you plot a function like sechx, cos(lnx), etc with a scientific calculator?
Do you need to actually rearrange to see if its invertible or is there a other method? (aka just by looking?)

a) sechx is not 1-1 (can tell from plotting it)

b) Not sure

c) not 1-1 (from graph)
 
  • #5
adelaide87 said:
Quick question - how do you plot a function like sechx, cos(lnx), etc with a scientific calculator?
Do you need to actually rearrange to see if its invertible or is there a other method? (aka just by looking?)

a) sechx is not 1-1 (can tell from plotting it)

b) Not sure

c) not 1-1 (from graph)

I'm not much on calculators so I can't answer the first question, but if you plotted sech(x) then, yes, it's not invertible, BUT you are only looking at the (0,infinity) part. Do you want to rethink that opinion? To prove a function is NOT invertible you only need to find two values of x, say x1 and x2 such that f(x1)=f(x2).
 
  • #6
Indeed, with sech(x), you need to look only at the positive numbers. Be careful there.

Also, cos(ln(x)) is easy to do on a TI calculator. Just make sure you know your bounds.
 

1. Are all matrices invertible?

No, not all matrices are invertible. A matrix can only be inverted if it is a square matrix and if its determinant is non-zero.

2. How can I determine if a matrix is invertible?

You can determine if a matrix is invertible by calculating its determinant. If the determinant is non-zero, then the matrix is invertible. If the determinant is zero, then the matrix is not invertible.

3. What is the significance of a matrix being invertible?

A matrix being invertible means that it has an inverse matrix, which can be used to solve linear equations and perform other operations on the matrix.

4. Can a non-square matrix be invertible?

No, a non-square matrix cannot be invertible because it does not have an equal number of rows and columns, which is a requirement for a matrix to have an inverse.

5. Can a matrix have more than one inverse?

No, a matrix can only have one inverse. If a matrix has more than one inverse, then it is not considered to be invertible.

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