## Composite functions

A composite function is a function created by exchanging one function into another function.

A composite function is a function created by exchanging one function into another function.

Let $$X$$ and $$Y$$ be two sets, and $$f$$ be a one-to-one function defined by $$f:{X} \rightarrow Y$$ with domain $$X$$ and range $$Y$$. Then, $$f^{-1}:{Y} \rightarrow X$$ is called the inverse function. Here, $$f^{-1}\left ( x\right )$$ does not mean $$\frac{1}{f\left (x \right )}$$.

A function can be thought of as two sets. Let $$X$$ and $$Y$$ be two sets and $$f$$ be the function from the set $$X$$ to $$Y$ then the function $$f$$ is defined as $$ f:X\rightarrow Y$$. Here, the set $$X$$ is called the domain or the set of inputs and set $$Y$$ is the range or the set of outputs. The function $$f$$ associated with each element is $$x$$ in $$X$$ with one and only element $$y$$ in $$Y$$.

Proportion is a statement that defines two specified ratios are equal to each other. Ratio and proportion are explained using fractions. If a fraction is expressed as $$a: b$$, it is called a ratio. On the other hand, when two ratios are equal, it is called a proportion.

A sequence is a set of numbers that usually follows a specific pattern in mathematics. Each element in the sequence is called an item.

A sequence in which every term is obtained by multiplying or dividing a particular number with the preceding number is a geometric sequence. For example, the geometric progression, $$5, 10, 15, 20, 25, \cdots$$, has a common ratio of $$5$$.

A series of numbers is in harmonic sequence if the reciprocals of all the sequence elements form an arithmetic sequence, which does not contain $$0$$. In a harmonic sequence, any terms in the sequence are considered the harmonic means of its two neighbours.

If in an expression, two expressions or values are written equal, then it is called an equation. For instance, $$3x+4y=12$$. On contrary to this, if an expression shows the relation between the two values with the sign ‘$$<$$’(less than), ‘$$>$$’ (greater than), ‘$$\leq$$’(less than or equal to), ‘$$\geq$$’(greater than or equal to), then it is termed as an inequality.

Graphical representation plays an important role in mathematics as it makes the overall material easy to understand for everyone. Either they are linear or quadratic, equation or inequalities, they can be shown graphically.

A linear equation can be written in different ways. An equation containing two variables x and y is said to form a linear equation in two variables. The highest degree of both the variables x and y in the equation is 1.

Simultaneous quadratics contains two equations, one linear equation and one quadratic equation. Any equation whose highest degree is 2 is called a quadratic equation.

The solutions of the quadratic equation are called zero or the root of the equation. For quadratic equation, the equation has two roots or zeros.

A quadratic equation is a polynomial equation of degree of order 2. ax^2 + bx + c is the general form of a quadratic polynomial. If we equate this polynomial to zero, we get a quadratic equation. The general form of a quadratic equation is ax^2 + bx + c = 0.

A simultaneous equation is where two algebraic expressions (typically in terms of x and y) intersect with each other. When you solve for a simultaneous equation, you are solving for both x and y. And they are co-ordinates.

In algebra, a polynomial consisting of variables and coefficients having the highest degree value of 2 is termed as a quadratic polynomial. The general form of a quadratic polynomial is ax^2+bx+c, where a, b and c are real numbers.

The factorisation process is the opposite of expanding the bracket. An algebraic expression for factorising means put the expression into the brackets by taking the common factors.

Factorising means the breaking or the decomposition of any entity into the product of other entities. Factorising is the process of finding the factors. It is like dividing an expression into a multiplication of relevant expressions. It is the reverse of expanding.

Brackets are the most commonly used symbols, such as the parentheses in an algebraic expression, to establish groups or explain the order of the operations to be performed.

A bracket is a symbol which helps us maintain the difference between two terms. Whenever we have to differ one mathematical term from another mathematical term, we use a bracket. For easy understanding, consider a bracket as a wall, a wall parting two rooms or houses.

An algebraic fraction is one in which the numerator and the denominator are both algebraic expressions.

An algebraic fraction is a fraction whose numerator and denominator are algebraic expressions.

Rearranging complicated formulae can be considered the peak of algebra skills. If you manipulate an equation and make your required variable the subject before adding the numeric values, solving for variables becomes easy.

The ability to rearrange formulae or rewrite them is an important skill. There are four main things in rearranging a formula: Add or subtract the same quantity to both sides, and multiply or divide both sides by the same quantity.

An index is used to show how many times a number is multiplied by itself. The plural word for index is indices. If some number is raised to some power, then the power it is raised to is the index of that number.

The basic idea behind substitution is how we can put values of variables in the equation to find the values of other variables.

Substitution is the procedure of putting one thing like a number, a letter or symbol in place of other. In simple words, whenever we use numbers at the position of letters, it is called the substitution process.

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