I'm stuck on something in Discrete Maths, and it wasn't properly taught to us – and I can't really find anything greatly useful in our textbook about it.
I'm given set $A = \left\{ x \in [0,\infty) | 3.2 < x < 8.9 \right\}$. So, the set consists of all Real-positive numbers between (not including) 3.2 and 8.9. I'm tasked with finding the pre-image of the set using the function $g(x) = \lceil x^2 \rceil$. What are the steps to finding the preimage of A, represented by $g^{-1}(A)$?
Our teacher glossed over it pretty quickly with one example:
$h(x) = \lfloor 3x−1 \rfloor$. Let $A = \left\{x \in R | 4 < x < 10 \right\}$. She picked the numbers $h^{-1}(5, 6, 7, 8, 9)$ and jumped to the answer of $[2, 11/3)$. I'm guessing she picked 5-9 because they're in the range of $4 < x < 10$; and the 2 and 11/3 came from setting those numbers equal to the function $3x-1$. She didn't explain this process at all, so that's entirely conjecture.
The answer to my original problem is $[-\sqrt{8}, -\sqrt{3}) U (\sqrt{3}, \sqrt{8}]$. Could anyone help me work through this example? Which numbers would I use in the function to find these values? Would they just be 3 and 8, set equal to $x^2$, to get the square root values? Why were the square roots of 8 included, but not 3? Because it wouldn't be between 3.2 and 8.9? Where does the ceiling function (choosing the highest integer) come into the equation?
Wouldn't 3.2 equal 4 and 8.9 equal 9? I'm sorry, I'm just really confused, and I'm rambling trying to figure something out.
Any help or shove in the right direction would be hugely appreciated. Thank you!
Best Answer
I would recommend thinking of your functions as composite functions:
$g(x) = \lceil x^2 \rceil = g_2(g_1(x))$, where $g_1(x) = x^2$ and $g_2(x) = \lceil x \rceil$.
To find the preimage, we work backwards. First, we find the preimage $g_2^{-1}(A) = (3,8]$.
The reason 3 isn't in that preimage, is because the ceiling of 3 is 3, which isn't in $A$. Similarly, the reason that the preimage only goes up to 8, is because for any number larger than 8, say 8.01, the ceiling is 9, which isn't in $A$.
Now that we have the preimage $g_2^{-1}(A) = (3,8]$, we can find the preimage preimage $g_1^{-1}\big( (3,8] \big)$, which will be the answer to the overall question.
To find that preimage, we can break up the domain into two regions (and I'm assuming that the domain of the function isn't given, and is assumed to be the largest possible subset of $\mathbb{R}$ on which the expression is well defined, which in this case is all of $\mathbb{R}$).
On the region $x \in [0,\infty)$, since $g_1$ is continuous and strictly increasing and $(3,8]$ is an interval, the preimage is:
$g_1^{-1}\big( (3,8] \big) = ( g_1^{-1}(3), g_1^{-1}(8)] = ( \sqrt{3}, \sqrt{8}]$.
Note that in the above equations, the first $g_1^{-1}$ stands for "preimage", but in the center the two $g_1^{-1}$ stand for the inverse of the function, which is the positive square root, since we're in the region $x \in [0,\infty)$.
On the region $x \in (-\infty, 0)$, since $g_1$ is continuous and strictly decreasing and $(3,8]$ is an interval, the preimage is:
$g_1^{-1}\big( (3,8] \big) = [ g_1^{-1}(8), g_1^{-1}(3)) = [ -\sqrt{8}, -\sqrt{3})$.
The overall preimage is the union of these, which is the answer that you already know.
Hopefully this clears up the process.
Applying the same technique to the example that your teacher did quickly:
$h_2(h_1(x)) = \lfloor 3x-1 \rfloor$, with $h_1(x) = 3x-1$ and $h_2(x) = \lfloor x \rfloor$.
$h_2^{-1}\big( (4,10) \big) = [5,10)$.
That's the trickiest part, so let me explain why. Since we're dealing with the floor function this time, if $x \in (4,5)$ then it will be mapped to 4, which isn't in $(4,10)$. So our preimage starts at 5. On the other hand, if $x \in (9,10)$, then it will be mapped to 9, which is in $(4,10)$, so our preimage can go all the way up to, but not including, 10.
The next step is easier than in your problem, since $h_1$ is continuous and strictly increasing on all of $\mathbb{R}$:
$h_1^{-1}\big( [5, 10) \big) = \big[ h_1^{-1}(5), h_1^{-1}(10) \big) = \big[ 2, 11/3 \big)$
If you ever have trouble reasoning through the preimage as I've done above, a slower but less error prone way to find the preimage is to graph the function, then mark on the y axis the set that you want to find the preimage for, and use the graph to see what set on the x axis is the preimage of that.