Let \(f:R→R\) be defined by \(f(x)=\){\(2x+3,x≤5 3x+α,x>5\) .Then the value of \(α\) so that f is continuous on \(R\) is
Let \(f:R→R\) be defined by \(f(x)=\){\(2x+3,x≤5 3x+α,x>5\) .Then the value of \(α\) so that f is continuous on \(R\) is
\(2\)
\(-2\)
\(3\)
\(-3\)
\(0\)
The Correct Option is B
Approach Solution - 1
Step 1: For continuity at \( x = 5 \), the left-hand and right-hand limits must equal \( f(5) \).
Step 2: Compute left-hand limit (\( x \to 5^- \)): \[ \lim_{x\to5^-} f(x) = 2(5) + 3 = 13 \]
Step 3: Compute right-hand limit (\( x \to 5^+ \)): \[ \lim_{x\to5^+} f(x) = 3(5) + a = 15 + a \]
Step 4: Set them equal for continuity: \[ 13 = 15 + a \] \[ a = -2 \]
Conclusion: The value of \( a \) that makes \( f \) continuous is \(\boxed{B}\) (\(-2\)).
Approach Solution -2
Given
Let \(f:R→R\) be defined by \(f(x)=\){\(2x+3,x≤5 3x+α,x>5\) .
Then the value of \(α\) so that f is continuous on \(R\) is
\(f ( 5^−) = 13 = 15 + α\)
\(α = 13 − 15\)
\(α = −2\)
Top Questions on Relations and functions
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
- JEE Main - 2025
- Mathematics
- Relations and functions
- Let $f: A \to B$ be defined by $f(x) = \frac{x - 2}{x - 3}$, where $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Discuss the bijectivity of the function.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- Let \( R \) be a relation defined on a set \( \mathbb{N} \) of natural numbers such that \( R = \{(x, y) : xy \text{ is a square of a natural number}, x, y \in \mathbb{N} \} \). Determine if the relation \( R \) is an equivalence relation.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- Show that the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 4x^3 - 5 \), \( \forall x \in \mathbb{R} \), is one-one and onto.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- If $f(x) = -2x^2$, then the correct statement is:
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
Questions Asked in KEAM exam
- Benzene when treated with Br\(_2\) in the presence of FeBr\(_3\), gives 1,4-dibromobenzene and 1,2-dibromobenzene. Which type of reaction is this?
- KEAM - 2025
- Haloalkanes and Haloarenes
- 10 g of (90 percent pure) \( \text{CaCO}_3 \), treated with excess of HCl, gives what mass of \( \text{CO}_2 \)?
- KEAM - 2025
- Stoichiometry and Stoichiometric Calculations
- Evaluate the following limit: $ \lim_{x \to 0} \frac{1 + \cos(4x)}{\tan(x)} $
- KEAM - 2025
- Limits
- Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b} \), \( |\mathbf{a} + \mathbf{b}| = \sqrt{29} \), \( \mathbf{a} \cdot \mathbf{b} = ? \)
- KEAM - 2025
- Vector Algebra
- In an equilateral triangle with each side having resistance \( R \), what is the effective resistance between two sides?
- KEAM - 2025
- Combination of Resistors - Series and Parallel
Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
