GMAT Quant: Algebra – Functions

The 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, then you must know that this question is a functions question.

But what does it really mean?

It means that “whatever goes within the brackets with x, the same will appear on the other hand of the equation.” So if you have 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)=1x

f(x)=x√

f(x)=1+x3√x35–5√x13

If any polynomial function y=p(x) is an algebraic function, then, y–p(x)=0

If any rational function y=p(x)q(x), then q(x)y–p(x)=0

The nth root of any polynomial y=p(x)‾‾‾‾√n is an algebraic function, solving the equation yn–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)

0

1

2

5

7

Solution:

2 (3x – 3) – 3 = 3 (3x – 3) – 3

6x – 6 = 9x – 9

x = 1

Answer: 2

Question :

For which of the following functions f is f(x) = f(1-x) for all x?

(A) f(x) = 1 – x

(B) f(x) = 1 – x^2

(C) f(x) = x^2 – (1 – x)^2

(D) f(x) = x^2*(1 – x)^2

(E) f(x) = x/(1 – x)

Solution: What does this mean: f(x) = f(1-x)? It means that given a certain expression in x called f(x), for which function will that be the same as f(1-x) i.e. when you substitute x by (1-x), which expression will stay the same? Let’s look at each option:

(A) f(x) = 1 – x

Substitute (1 – x) in place of x to see what f(1 – x) looks like.

f(1 – x) = 1 – (1 – x)

f(1 – x) = x

f(x) is not the same as f(1-x) here. Ignore this option.

(B) f(x) = 1 – x^2

Substitute (1 – x) in place of x to see what f(1 – x) looks like.

f(1 – x) = 1 – (1 – x)^2

f(1 – x) = 2x – x^2

f(x) is not the same as f(1-x) here. Ignore this option.

(C) f(x) = x^2 – (1 – x)^2

Substitute (1 – x) in place of x to see what f(1 – x) looks like.

f(1 – x) = (1 – x)^2 – (1 – (1-x))^2

f(1 – x) = (1 – x)^2 – x^2

f(1 – x) = -x^2 + (1 – x)^2

f(x) is not the same as f(1-x) here. Ignore this option.

(D) f(x) = x^2*(1 – x)^2

Substitute (1 – x) in place of x to see what f(1 – x) looks like.

f(1 – x) = (1 – x)^2 * (1 – (1 – x))^2

f(1 – x) = (1 – x)^2 * x^2

f(1 – x) = x^2 * (1 – x)^2

Note that here, f(x) = f(1 – x), so this must be our answer. Still, let’s take a look at (E) as well for practice.

(E) f(x) = x/(1 – x)

Substitute (1 – x) in place of x to see what f(1 – x) looks like.

f(1 – x) = (1 – x)/(1 – (1-x))

f(1 – x) = (1 – x)/x

f(x) is not the same as f(1-x). Ignore this option.

Answer (D)

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