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FAQs on Important Formulas: Relations and Functions - Mathematics (Maths) for JEE Main & Advanced
| 1. What are the types of relations in mathematics? |  |
Ans. The types of relations in mathematics include:
1. <b>Reflexive Relation</b>: A relation R on a set A is reflexive if for every element a in A, (a, a) ∈ R.
2. <b>Symmetric Relation</b>: A relation R is symmetric if for every a, b in A, if (a, b) ∈ R, then (b, a) ∈ R.
3. <b>Transitive Relation</b>: A relation R is transitive if for every a, b, c in A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
4. <b>Antisymmetric Relation</b>: A relation R is antisymmetric if for every a, b in A, if (a, b) ∈ R and (b, a) ∈ R, then a = b.
5. <b>Equivalence Relation</b>: A relation that is reflexive, symmetric, and transitive.
| 2. How do you determine if a function is one-to-one (injective)? | |
Ans. A function f: A → B is one-to-one (injective) if for every pair of distinct elements a1 and a2 in A, f(a1) ≠ f(a2). This means that no two different inputs have the same output. To verify injectivity, you can use the contrapositive method: if f(a1) = f(a2), then a1 must equal a2.
| 3. What is the difference between a function and a relation? | |
Ans. A relation is a set of ordered pairs, where the first element is related to the second element in some way. A function is a special type of relation where each input (from the domain) is associated with exactly one output (in the codomain). In other words, in a function, no two ordered pairs can have the same first element with different second elements.
| 4. What are the different types of functions based on their characteristics? | |
Ans. The different types of functions based on their characteristics include:
1. <b>Linear Function</b>: A function of the form f(x) = mx + b, where m and b are constants.
2. <b>Quadratic Function</b>: A function of the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.
3. <b>Polynomial Function</b>: A function that is a polynomial of degree n, represented as f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0.
4. <b>Rational Function</b>: A function represented as the ratio of two polynomials.
5. <b>Exponential Function</b>: A function of the form f(x) = a*b^x, where a is a constant and b > 0.
| 5. How do you find the inverse of a function? | |
Ans. To find the inverse of a function f(x), follow these steps:
1. Replace f(x) with y: y = f(x).
2. Swap x and y: x = f(y).
3. Solve for y in terms of x to express it as y = f⁻¹(x).
4. The inverse function exists if f is bijective (both injective and surjective).
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