Algebra of Functions, Rational, Trigonometric

Algebra of Functions, Rational, Trigonometric

Algebra is a branch of mathematics that comprises building equations with variables to solve numerical problems. Algebra includes real numbers, complex numbers, vectors, matrices, and other mathematical forms.

Algebraic Functions

In addition to fractional or rational exponents, algebraic operations such as subtraction, addition, multiplication, and division are employed in an algebraic function. 

Consider an algebraic function to be a machine that accepts real numbers, performs mathematical operations, and gives an output.

The integers that go into an algebraic function are referred to as the input x or domain. A function can take any number as long as it isn’t divided by zero or produces a negative square root. As long as the range is one value for each domain used, a function can do several mathematical operations with it. The numbers that come out of a function are referred to as the output y, or range. Keep in mind that one value in equals one value out.

Algebraic functions include linear, quadratic, cubic, polynomial, rational, and radical equations, among others.

• Tables

A table can show us if there is only one domain and one range, which is one way of recognizing an algebraic function. y=x + 2 is an example of a function that adds to the domain to get the range. y=3x is an example of a function that multiplies the domain to get the range.

• Graphs

When it comes down to identifying functions, we can use graphs by showing pairs onto a Cartesian System, where the x-values are on the horizontal line and the y-values are on the vertical line. At the intersection of the ordered pairs (different input values (x) with corresponding output values (y)), the point is graphed.

• Vertical Line Test

If a graph passes the vertical line test, it is a function. In this test, a vertical line drawn anywhere on a graph will only cross in one spot. The one-in-one-out rule is violated when a vertical line crosses two locations on a graph.

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomial functions. The coefficients of the polynomials, as well as the values taken by the function are not always rational.

Any function with only one variable x, is referred to as a rational function if and only if it can be represented as:

F (x)= P(x)/Q(x) and Q(x)≠0

What are asymptotes?

An asymptote is a line that a curve tends to as it extends till infinity to never actually touch it. 

Let’s discuss the kinds of asymptotes. 

• Vertical Asymptotes

The vertical asymptotes of a rational function may be found by finding the factors of the denominator that are not common to the factors of the numerator. Vertical asymptotes occur at the zeros of such factors.

• Horizontal Asymptotes

Horizontal asymptotes help describe the behaviour of a graph as the input becomes extremely large or very small, whereas vertical asymptotes help define the graph behaviour as the output becomes either very large or very small. Remember that the end behaviour of a polynomial will be the same as the leading term. Similarly, the end behaviour of a rational function will be the same as the ratio of the numerator and denominator functions leading terms.

• Oblique Asymptotes

Oblique asymptotes occur when the order of the denominator of a rational function is one less than the order of the numerator.

The Domain of a Rational Function

The domain is a set of all values that x in a function f(x) can take.

Let’s understand this with the help of some examples.

Example 1.

f(x)=x/x-3

The denominator has only one factor, x cannot be equal to 3. As a result, the domain of f is defined as the set of all numbers other than 3.

Trigonometric Functions: 

Circular functions, sometimes known as trigonometric functions, can be simply described as functions of a triangle’s angle. This means that these trigonometric functions determine the relationship between the angles and sides of a triangle. Cosecant, sine, cosine, secant, tangent, and cotangent are the basic trigonometric functions. 

Here you can also learn about trigonometric identities.

Some trigonometric formulae and identities help find the angles of the triangle and signify the relationship between the functions. All of these trigonometric functions, along with their formulae, are presented in detail here to help you comprehend them.

Trigonometric Functions

The sine, cosine, and tangent angles are the fundamental classifications of trigonometric functions. 

• Sine Function

The ratio of the opposite side length of an angle to the hypotenuse is the sine function of an angle. 

Sin A =Opposite side length/Hypotenuse  =Perpendicular/Hypotenuse

Opposite side length is called perpendicular.

• Cos Function

The cosine of an angle is the ratio of the neighbouring side’s length to the hypotenuse’s length. The cos function can be calculated using the below formula.

Cos A = Adjacent side length /Hypotenuse = Base/Hypotenuse

Adjacent side length is called base.

• Tan Function

The tangent function is the ratio of the opposing side’s length to the adjacent side’s length. It’s worth noting that  tan can also be expressed as the ratio of sine and cos. 

Tan A = Opposite/Adjacent = Perpendicular/Base

Also, in terms of sin and cos, tan can be represented as:

Tan A = sin A/cos A

• Cosecant, Cotangent, and Secant Functions

Cotangent is the proportion between adjacent and opposite sides. 

Secant is the ratio of the hypotenuse to the neighbouring side. 

The ratio of the hypotenuse to the opposite side (hypotenuse/perpendicular) is called cosecant.