Last week, we looked at the basics of how to handle function questions. Today, let’s look at a couple of questions. We will start with an easier one and then go on to a slightly tougher one.
Question 1: If f(x) = 343/x^3, what is the value of f(7x)* f(x/7) in terms of f(x)?
Solution: We discussed that to get f(a) given f(x), all you need to do is substitute x with a.
f(x) = 343/x^3
f(7x) = 343/(7x)^3 = 1/x^3
f(x/7) = 343/(x/7)^3 = 343*343/x^3
So we get f(7x) * f(x/7) = (1/x^3) * (343*343/x^3) = (343/x^3)^2
But we know that 343/x^3 = f(x)
So, f(7x) * f(x/7) = (f(x))^2
There are other ways of solving this too:
Say x = 1, then f(1) = 343
f(7x)* f(x/7) = f(7)*f(1/7) = (343/7^3) * (343/(1/7)^3 = (343)^2
So f(7x)* f(x/7) = (f(1))^2
We hope you see that the question was not difficult to solve. Once you get over your fear of symbols, it is quite straight forward.
Now, let’s take a question similar to an official question from the GMAT paper tests:
Question 2: The function f is defined for each positive three-digit integer n by f(n) = 2^x * 3^y * 5^z, where x, y and z are the hundreds, tens, and units digits of n, respectively. If m and v are three-digit positive integers such that f(m) = 25f(v), then m-v=?
Solution: The question may seem a bit difficult to understand first so let’s take one sentence at a time:
“f is defined for each positive three-digit integer n by f(n) = 2^x * 3^y * 5^z, where x, y and z are the hundreds, tens, and units digits of n”
Let’s take an example to see how to make sense of it: say 146 is a three digit positive integer. So f(146) = 2^1 * 3^4 * 5^6
In the same way, f(283) = 2^2 * 3^8 * 5^3
“If m and v are three-digit positive integers such that f(m) = 25f(v)”
So f(m) = 5^2 * f(v)
If m is represented as abc and v as def, then (2^a * 3^b * 5^c) = 5^2 * (2^d * 3^e * 5^f)
Note that for the left hand side to be equal to right hand side, a = d, b = e and c = 2 + f.
So the units digit of m is 2 more than the units digit of v but their tens and hundreds digits are the same.
So m – v = 2.
If you are still not sure how we arrived at this, take an example.
f(m) = f(266) = 2^2 * 3^6 * 5^6
f(v) = f(264) = 2^2 * 3^6 * 5^4
The difference between f(m) and f(v) will be of 5^2 since their units digits are 2 away from each other.
Hope next time you see a functions question, you will not get spooked and will, instead, take it in your stride!
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!