1. ## 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.

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

Could someone please explain it to me how to do it? Thanks in advance.

2. ## Absolute values and inverses

Hello Superman-Prime
Originally Posted by Superman-Prime
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.

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?

Hello Superman-PrimeWelcome to Math Help Forum!
Thanks

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?