Common Functions

In Mathematics, functions are defined as an expression, rule, or law that defines the relationship between one variable known as the independent variable and another variable known as a dependent variable). The relationships are commonly represented as y = f(x). In addition to f(x), other common symbols such as g(x) and p(x) are often used to represent the functions of an independent variable x, specifically when the nature of function is unspecified or unknown.

In this article, we will discuss some common functions such as common trig functions, common square, common trig values, common trig angles, common roots of quadratic equations, meaning of gcd, gcd and lcm of polynomials etc.

Let us begin with the meaning of GCD.

Meaning of GCD

In Mathematics, the GCD (also known as greatest common factor) of two numbers is the largest number that divides both of them.  For example, the greatest common divisor of 8 and 12 is 4, as 4 divides both 8 and 12 and no other larger number has this property. This concept can be easily extended to a set of two or more numbers. The greatest common divisor (gcd) of the set of two numbers is the largest number that divides each of them.

The gcd is commonly represented as gcd(x,y), or when the term is clear simply (x,y).

GCD OF Negative Numbers

GCD of negative numbers is calculated similarly to the gcd of positive numbers. For example, the gcd of 18 and - 4 is 2, as no other larger number has this property. The negative sign in -4 can be ignored as this will not affect divisibility.

GCD of Polynomial

The GCD of two polynomials is the polynomial of the highest degree that divides both the polynomials. If the polynomial a(x) is the gcd of two numbers p(x) and q(x), we can write this as:

Gcd a(x) = gcd p(x) , q(x))

To find the gcd of two polynomials using factoring, simply find the factors of two polynomials. Then, find the product of all common factors.

GCD of Polynomial Example

Find the gcd of the polynomial 21a²b and 35ab².

Solution:

Step 1:

To find the gcd of 21a²b and 35ab², we will first find its factors using the prime factorization method.

Factors of 21a²b = 3 × 7 × a × a × b

Factors of 35ab² = 5 × 7 × a × b × b

Step 2:

Further, we  will find the common factors of a given polynomial.

Common factors of 21a²b and 35ab² = 7 × a × b

Therefore, gcd of 21a²b and 35ab² is 7ab.

LCM of Polynomial

LCM (also known as least common multiple) of two or more polynomials is defined as the polynomial of the lowest degree, having a smaller numerical coefficient that is exactly divisible by the given polynomials. The coefficients of the highest degree terms of the given polynomial retain the signs similar to the coefficients of the highest degree in the product.

To find the LCM of each set of polynomials given below, we will first find the factors of each polynomial by the method of factorization, and then multiply the factors with the highest exponent.

Example:

Find the LCM of each set of polynomials:

1. 16a, 8a²b³. 5a³b

Solutions:

Factors of 16a = 2 × 2 × 2 × 2 × a

Factors of 8a²b³ = 2 × 2 × 2 × a × a × b × b × b

Factors of 5a³b = 5 × a × a × a × b

LCM = 2 × 2 × 2 × 2 × 5 × a × a × a × b × b × b

 = 80a³b³

Hence, the required LCM is 80a³b².

2. 7p², 9pq³, 21pqr⁴

Factors of 7p² = 7 × p × p

Factors of 9pq³ = 3 × 3 × p × q × q × q

Factors of 21pqr⁴ = 3 × 7 × p × q × r × r × r × r

LCM = 7 × 3 × 3 × p × p × r  × r × r × r

= 63p²q³r⁴

Hence, the required LCM is 63p²q³r⁴.

Quadratic Function

A quadratic function is a function that can be rearranged in standard form as ax² + bx + c = 0. Here ‘x’ represents the unknown numbers whereas a, and b represents the known numbers.  Also a ≠ 0.

Common Roots of Quadratic Functions Conditions

A root of the quadratic equation  ax² + bx + c = 0 is a real or complex number, say , which satisfies the equation that is a² + b + c = 0. The root of the quadratic equation ax² + bx + c = 0, with a ≠ 0 is derived as:

The following are the conditions of the common root of quadratic equation.

• Determining the conditions that the equations a₁x² + b₁x + c₁ = 0 and  a₂x² + b₂x + c₁ = 0 comprises at least one common root or both the roots common.

• At least one common of the quadratic equation: (c₁a₂ - c₂a₁)/(a₂b₂ - a₂b₁) = (b₁a₂ - b₂a₁)/(c₁b₂ - c₂b₁).

• Both common roots of quadratic equations a₁/a₂ = b₁/b₂ = c₂/c₂.

Example

Find the value of such that the equation x² + αx + α + 2 = 0 and x² + ( 1 - α)x + 3 - α = 0 have exactly one common root.

Using the conditions of common roots (c₁a₂ - c - 2a₁)² = (a₁b₂ - a₂b₁)(b₂c₁ - b₁c₂)

We have,

[(α + 2) × 1 - 1 × (3 - α)²] = [α.(3 - α) - (α + 2)(1 - α)][( 1 - α). 1 - α.1]

(2α - 1)²  = 2(1 - 2α)(2α + 1)

(2α - 1)² = 0

α = 1/2

By substituting the value = 1/2 in the given equation x² + αx + α + 2 = 0 and x² + (1 - α)x + 3 - α = 0, we get 2 equations as 2x²+ x + 5 = 0 and 2x²+ x + 5 = 0. Here, we can see that two equations are similar i.e. both the equations will have both the roots in common . Therefore, α = 1/2, And, there is no such values of for which the equations x² + αx + α + 2 = 0 and x² + (1 - α)x + 3 - α = 0 has exactly one common root.

Common Trigonometric Functions

Trigonometry is derived from two words trignon (or triangle), and metria (or measure). It is the study of the relation between the angles and sides of a triangle. Here, we will explore the common trig functions that connect the measures of the angles with the length of their sides.

The trigonometric functions relate the angles in a right-angled triangle and the ratio of their sides. The most widely used trigonometry functions are the sine, the cosine, and the tangent ( abbreviated as sin, cos, and tan). The reciprocals are respectively the cosecant, the secant, and the cotangent ( abbreviated as csc, sec, and cot).

In the above right-angled triangle, if we consider the angle θ, and represents the side with θ, then ‘a’ represents the length of the adjacent side, ‘b’ represents the length of the opposite side, and ‘c’ represents the length of the hypotenuse side. Then the common trig functions can be expressed as:

Sin θ = Opposite Side/Hypotenuse Side = b/c

Cos θ = Adjacent Side/Hypotenuse Side = a/c

Tan θ = Opposite Side/Adjacent Side = b/a

Cosecant θ = Hypotenuse Side/Opposite Side = c/b

Secant θ = Hypotenuse  Side/Adjacent Side = c/a

Cotangent θ = Adjacent Side/Opposite Side = a/c

Common Trig Angles

The angles in which trigonometric functions are represented are known as trigonometry angles. The most common trig angles are 0°,30°,45°,60°, and 90°. These are the common trig angles of trigonometry ratios such as sine, cosine, tangent, secant, cosecant, and cotangent.

The trigonometric values of common trig functions such as sine, cosine, tangent, secant, 2cosecant, and cotangent, deals with the measurement of the length of the sides and angles of a right-angled triangle. The common trig values for common trig angles are represented below in tabular format.

Common Trig Values

Here are some common Functions that are widely used and tier graphs:

Common Functions and Graphs

Linear Function	f(x) = mx + b
Square Function	f(x) = x2
Cube Function	f(x) = x3
Square root Function	f(x) = √x
Sine Function
Cosine Function
Tangent Function
Absolute Value Function	f(x) = |x|
Reciprocal Function	f(x) = 1/x
Logarithmic Function	f(x) = ln(x)
Exponential Function	f(x) = ex
Quadratic Function

Solved Example

1. Find the value of sin 35°, cos 35°,and tan 35°.

Solution:

Sin 35° = Opposite Side/Hypotenuse Side = 2.8/4.9 = 0.57

Cos 35° = Adjacent Side/Hypotenuse Side = 4.0/4.9 = 0.82

Tan 35° = Opposite Side/Adjacent Side = 2.8/4.0 = 0.70

2. Find the discriminant of the quadratic function f(x) =x² - 3x + 4

Solution:

The given quadratic function is f(x) =x² - 3x + 4

On comparing the given quadratic function with ax² + bx + c = 0, we get the value of a = 1, b = -3, and c = 4

Discriminant of Quadratic Functions = b² - 4ac

= 3² - 4(1) (-4)

= 9 + 16 

= 25

Therefore, the discriminant is 25.

3. Find the GCD of 48 and 180.

Solution:

Step 1: Finding factors of 48 and 140 using prime factorization method.

Factors of 48 = 2 × 2 × 2 × 2

Factors of 140 = 2 × 2 × 3 × 3 × 5

Step 2: Finding common factors of 48 and 140.

Common factors of 48 and 140 are 2 × 2 × 3.

GCD = 12

Therefore, the greatest common divisor of 48 and 140 is 12, as no other large number can divide these numbers completely.