Algebra

Functions:

Have you heard dependent variables and independent variables in economics? There are some variables that are dependent and depends on other variables that are independent or may be dependent again. For instance Price is dependent variable and depends on demand, which is independent. (Keeping other-things remain same.)

Similarly, in functions, one variable is dependent, and other is independent.

For instance,

y is a function of x. Mathematically,

ƒ(x) = y

Where value of y depends on value of x, so y is a function of x.

Scenario 1:

Suppose, if ƒ(x) = x² —————— (eq. 1)

and ƒ(k) = 64

What is k?

Solution:

Always remember function is like a machine. Where you put some input, and the machine gives you output. So

ƒ(x) = x²
Here x is input, and x² would be output. i.e when you put some input, the machine squares the input, and give you output.

So if the input is k, it result would be k².

Let’s put k = x in equation in (eq. 1)

⇒ ƒ(k) = k²

Given that,

ƒ(k) = 64

⇒ k² = 64

⇒ k = ±8 Answer

Remember that functions are of two categories:

1. Linear function

2. Non-linear or Quadratic function

1. Linear function:

Linear functions are those that represents equation of a line. In other words a function in which a variable has only 1 exponent (power). For instance,

Let’s consider the following table:

As we know that

Slope = ^{(7 – 5)}⁄_{(2 – 1)} = 2

or

Slope = ^{(9 – 7)}⁄_{(3 – 2)} = 2

or

Slope = ^{(9 – 5)}⁄_{(3 – 1)} = 2

Now, let’s say if y-intercept (i.e c) = 3, then equation of that line (i.e linear function) is:

y = mx + c

⇒ y = 2x + 3

⇒ ƒ(x) = 2x + 3

I hope you understood similarity between a linear function and a linear equation (i.e equation of a line). Now, let’s learn in depth of linear function.

If, ƒ(1) = 4, and slope = –0.5 then what is value of ƒ(6)?

Solution:

You just need to remember the following steps that will help you in solving and finding any kind of linear function value.

Given that:

ƒ(1) = 4, and slope = –0.5

And we need to find value of ƒ(6). Just remember following:

ƒ(x) = ƒ(1) + (x – 1) × slope

⇒ ƒ(6) = 4 + (6 – 1) × (–0.5) = 4 + 5 × (–0.5) = 4 – 2.5 = 1.5 Answer

The above method is used only when value of ƒ(1) and slope is given. But what if the two values are not given and we need to find value of ƒ(1)? Let’s learn :)

Let’s say,

If ƒ(2) = 7 and ƒ(3) = 9, find value of ƒ(1)?

Solution:

As you know,

So,

m = Slope = ^{(9 – 7)}⁄_{(3 – 2)} = 2

Now, you know that:

ƒ(3) = ƒ(1) + (3 – 1) × m

⇒ 9 = ƒ(1) + 2 × 2

⇒ ƒ(1) = 9 – 4 = 5 Answer

You may check back to see same values that has been used from the previously discussed table in this lecture. And remember this question is solvable only if it is said that the function is a linear function. These type of hard scenario rarely tested and only tested when you start touching 99 percentile in your exam, as it happen to me twice. :D

Function as in independent variable (i.e function inside another function).

Non-linear functions:

Parabola and Hyperbola:

Inequalities and Absolute Values:

Inequalities:

As we know,

3 < 7

By multiply both sides by –2, inequality sign will flip,

⇒ –6 > –14

Similarly, in case of division with negative number, the flip of inequality sign occur.

Let’s learn something practical, and applicable in exams.

Scenario 1:

Given that:

–40 < x < 10

–20 < y < 60

Find the maximum values and minimum values of the following:

x + y = ?

x – y = ?

x × y = ?

^x⁄_y = ?

Please try this by your own first, and then go ahead for checking your skill. These questions will help broaden your mind and scope of thinking. Because you answer

x + y = Max x + Max y = 15 + 60 = 75 Answer

x – y = Max x – Min y = 15 – (–10) = 25 Answer

x × y = Max product of x and y = (–40 × –20) = 80 Answer

^x⁄_y = Max ratio of x to y = ¹⁰⁄₀ = ∞ Answer

Note that if we divide 10 by 0.000000000000000001, it will give a very big value. So if we divide 10 by 0, it will give undefined answer. Also note that x can be any number, so don’t assume only integer value of x. Never forget the definitions of numbers and integers, that you’ve studied in beginners study plan.

Now, let’s find minimum values:

x + y = Min x + Min y = (–40) + (–20) = –60 Answer

x – y = Min x – Max y = (–40) – 60 = –100 Answer

x × y = Min product of x and y = (–40) × 60 = –2400 Answer

^x⁄_y = Min ratio of x to y = ^–40⁄₀ = ∞ Answer

Note that if we divide –40 by 0.000000000000000001, it will give a very small number (i.e very big negative number like –1000000000000000000). So if we divide 10 by 0, it will give undefined answer.

Now, what if x and y are integers, and x & y ≠ 0?

Well, in that case, the maximum and minimum values of (x + y), (x – y) and (x × y) will remain same as find out above. But values of other will become as follows:

For maximum values:

^x⁄_y = Max ratio of x to y = ^–40⁄_–1 = 40 Answer

For minimum values:

^x⁄_y = Min ratio of x to y = ^–40⁄₁ = –40 Answer

Absolute Values:

Absolute value means non-negative value. It is usually calculated in terms of distance between two values etc, which cannot be negative. It is denoted by:

|x| = 4

⇒ x = ±4

But,

|4| = +4 only

You’ve seen that it works similar to square root of x², but only difference is that that it uses vertical bars instead of square root.

Also, similar to x², absolute values have following rule:

|x| ± |y| ≠ |x ± y|                         Similar to                         √x ± √y ≠ √x ± y

Now, let’s learn combination of both inequalities and absolute values.

If |x| > 4

⇒ Either   x > 4         or         x < –4

Note that and remember that, when absolute sign is removed, it results to two range of x in any inequality expression. And the sign of inequality will get flipped with the negative value, as show in above scenario with –4.