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Thread: Functions and Inverse

  1. #1
    Newbie Superman-Prime's Avatar
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    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.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    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 $\displaystyle g$ from having an inverse is if there are two distinct values of $\displaystyle x$ for which the values of $\displaystyle g(x)$ are the same. This can happen with things like $\displaystyle |x|$ because $\displaystyle x = 2$ and $\displaystyle x = -2$ (for example) both give $\displaystyle |x| = 2$.

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

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

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

    So $\displaystyle g(x) = 3 - 2x - 4= - 2x-1$

    OK?

    Grandad
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  3. #3
    Newbie Superman-Prime's Avatar
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    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 $\displaystyle g$ from having an inverse is if there are two distinct values of $\displaystyle x$ for which the values of $\displaystyle g(x)$ are the same. This can happen with things like $\displaystyle |x|$ because $\displaystyle x = 2$ and $\displaystyle x = -2$ (for example) both give $\displaystyle |x| = 2$.

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

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

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

    So $\displaystyle g(x) = 3 - 2x - 4= - 2x-1$

    OK?

    Grandad
    Understood fully. ThanX a lot!
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