Originally Posted by

**Grandad** 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