GMAT Quant: Algebra – Functions

Algebraic function can be defined as the root of a polynomial equation. These can be expressed in many different terms like algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.

What are Functions?

Functions are like machines, which takes an input and provides you with an output. When you witness equations of the form of f(x) =x3-5, just know that this question is of function.

But what it really means? It means that “whatever goes within the brackets with x, the same will appear on the other hand of the equation.” So ii you put x=2, then f(2) = 8 – 5 (you simply put the 2 wherever x appears in the function). The only most important point to keep in mind about functions is the working of functions. No matter how confusing or abstract they seem to be, the rule to follow is that, whatever is inside the parentheses and just plug it in wherever that variable appears on the other side of the equation.

Common examples of such functions are –

\(f(x) = \frac{1}{x}\)
\(f(x) = \sqrt{x}\)
\(f(x) = \frac{\sqrt{1 + x^{3}}}{x^{\frac{3}{5}} – \sqrt{5}x^{\frac{1}{3}}}\)

If any polynomial function \(y = p(x)\) is an algebraic function, then, \(y – p(x) = 0\)
If any rational function \(y =\frac{p(x)}{q(x)}\), then \(q (x) y – p (x) = 0\)
The nth root of any polynomial \(y = \sqrt[n]{p(x)}\) is an algebraic function, solving the equation \(y^{n} – p(x) = 0\)

Let’s solve a question and see how functions work.

Question: If f(x) = 3x-3,

For what value of x does

2 (f(x)) – 3 = f(3x -3)

  1. 0
  2. 1
  3. 2
  4. 5
  5. 7

Solution:

2 (3x – 3) – 3 = 3 (3x – 3) – 3

6x – 6 = 9x – 9

x = 1

Answer: 2

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