Results 1 to 3 of 3

Math Help - Functions and Inverse

  1. #1
    Newbie Superman-Prime's Avatar
    Joined
    Apr 2009
    Posts
    3

    Functions and Inverse

    Hello,

    I'm having trouble understanding this question related to absolute value function. I have all the answers, I just don't know how to calculate them.

    Q: A function g is defined by g:x |-> |2x-3| - 4 , for -2≤x≤k.

    i) State the largest value of k for which g has an inverse.
    ii) Given that g has an inverse, express g in the form g:x |-> ax+b , where a and b are constants.

    Answers:
    i) k=1.5
    ii) -2x-1

    Could someone please explain it to me how to do it? Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570

    Absolute values and inverses

    Hello Superman-Prime
    Quote Originally Posted by Superman-Prime View Post
    Hello,

    I'm having trouble understanding this question related to absolute value function. I have all the answers, I just don't know how to calculate them.

    Q: A function g is defined by g:x |-> |2x-3| - 4 , for -2≤x≤k.

    i) State the largest value of k for which g has an inverse.
    ii) Given that g has an inverse, express g in the form g:x |-> ax+b , where a and b are constants.

    Answers:
    i) k=1.5
    ii) -2x-1

    Could someone please explain it to me how to do it? Thanks in advance.
    Welcome to Math Help Forum!

    The thing that might prevent g from having an inverse is if there are two distinct values of x for which the values of g(x) are the same. This can happen with things like |x| because x = 2 and x = -2 (for example) both give |x| = 2.

    So with an expression like |2x-3| you need to look at the value of x that makes 2x - 3 = 0, because either side of that, you'll have two values of x giving the same value of |2x - 3|. Clearly that value is x=1.5. So when x = 1.5 + 1 (for example) and x = 1.5 - 1, the value of |2x - 3| is the same: it's 2 in each case.

    For part (i) we are told that x lies in the range -2 \le x \le k. So we are certainly allowed some values of x that are less than 1.5. Therefore if g is going to have an inverse, we must avoid any values of x that are greater than 1.5 in order to avoid repeating the values of |2x-3|. Hence k (the greatest permitted value of x) = 1.5.

    For part (ii) we know that g has an inverse. So we know that x \le 1.5, and hence 2x - 3 \le 0. So to find |2x - 3| we must be changing the sign of 2x - 3 from negative to positive. In other words, when x \le 1.5, |2x - 3| = -(2x-3) = 3 - 2x.

    So g(x) = 3 - 2x - 4= - 2x-1

    OK?

    Grandad
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie Superman-Prime's Avatar
    Joined
    Apr 2009
    Posts
    3
    Quote Originally Posted by Grandad View Post
    Hello Superman-PrimeWelcome to Math Help Forum!
    Thanks

    Quote Originally Posted by Grandad View Post
    The thing that might prevent g from having an inverse is if there are two distinct values of x for which the values of g(x) are the same. This can happen with things like |x| because x = 2 and x = -2 (for example) both give |x| = 2.

    So with an expression like |2x-3| you need to look at the value of x that makes 2x - 3 = 0, because either side of that, you'll have two values of x giving the same value of |2x - 3|. Clearly that value is x=1.5. So when x = 1.5 + 1 (for example) and x = 1.5 - 1, the value of |2x - 3| is the same: it's 2 in each case.

    For part (i) we are told that x lies in the range -2 \le x \le k. So we are certainly allowed some values of x that are less than 1.5. Therefore if g is going to have an inverse, we must avoid any values of x that are greater than 1.5 in order to avoid repeating the values of |2x-3|. Hence k (the greatest permitted value of x) = 1.5.

    For part (ii) we know that g has an inverse. So we know that x \le 1.5, and hence 2x - 3 \le 0. So to find |2x - 3| we must be changing the sign of 2x - 3 from negative to positive. In other words, when x \le 1.5, |2x - 3| = -(2x-3) = 3 - 2x.

    So g(x) = 3 - 2x - 4= - 2x-1

    OK?

    Grandad
    Understood fully. ThanX a lot!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. inverse functions
    Posted in the Algebra Forum
    Replies: 3
    Last Post: November 22nd 2009, 04:58 PM
  2. Replies: 2
    Last Post: October 19th 2009, 02:47 AM
  3. inverse one-one functions
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: September 12th 2009, 04:06 PM
  4. inverse functions
    Posted in the Algebra Forum
    Replies: 2
    Last Post: April 27th 2009, 07:54 AM
  5. are these 2 functions inverse of each other?
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: November 22nd 2008, 04:02 PM

Search Tags


/mathhelpforum @mathhelpforum