Core 3 functions question

Hello, having some trouble with part iii on this question. The mark scheme says "Attempt arrangement of f(x)=x or f^-1(x)=x" and "Apply discriminant to resulting quadratic equation", but I'm not sure how to get the quadratic equation in the first place.

Thanks in advance.

Hello, having some trouble with part iii on this question. The mark scheme says "Attempt arrangement of f(x)=x or f^-1(x)=x" and "Apply discriminant to resulting quadratic equation", but I'm not sure how to get the quadratic equation in the first place.

Thanks in advance.

Can you post all your thoughts and working for this question? It will be useful to see what you've tried and how far you got.


You should use the direct link to the image when using Imgur images on TSR otherwise it won't be displayed. So you should have used http://i.imgur.com/jpubfj1.png instead of http://imgur.com/a/fadLB.

I'm not too sure where to being really, I understand that the curves are reflections of each other through y=x meaning they wont meet. I'm unsure as to why you would use f(x)=x or f^-1(x)=x and also to get a quadratic equation by doing this. Sorry if this isn't making much sense, I'm not too sure about it myself.

Also apologies about the image situation, I'm new and still finding my way around here

Either would work but since you are given the function and not its inverse, you should be considering the equation $f(x) = x$.

Do you understand that if $f(x)$ and $f^{-1}(x)$ don't meet then $f(x) = x$ has no solutions?

$f(x) = x \Rightarrow \sqrt{mx+7} - 4 = x$

Move the 4 to the other side and then square both sides. You should end up with a quadratic after simplifying. Post your working if you get stuck.

$f(x) = x \Rightarrow \sqrt{mx+7} - 4 = x$
$f(x) = x \Rightarrow \sqrt{mx+7} = x + 4$
$f(x) = x \Rightarrow mx+7 = x^2 + 8x + 16$
$f(x) = x \Rightarrow mx = x^2 + 8x + 9$

Not sure how to simplify from here, if I divide by x then I won't have a quadratic anymore? The mark scheme says I need to get (m – 2)(m – 14) < 0.

$mx = x^2+8x+9$

$x^2+8x-mx+9=0$

$x^2+(8-m)x+9=0$

This is a quadratic with $a = 1, b = 8-m$ and $c = 9$. You need to find values of m where this quadratic has no solutions. You probably would have done questions like this at C1.