If
\(f(x) = \begin{cases} x + a, & x \leq 0 \\ |x - 4|, & x > 0 \end{cases}\) and \(g(x) = \begin{cases} x + 1, & x < 0 \\ (x - 4)^2 + b, & x \geq 0 \end{cases}\)
are continuous on R, then (gof) (2) + (fog) (–2) is equal to
If
\(f(x) = \begin{cases} x + a, & x \leq 0 \\ |x - 4|, & x > 0 \end{cases}\) and \(g(x) = \begin{cases} x + 1, & x < 0 \\ (x - 4)^2 + b, & x \geq 0 \end{cases}\)
are continuous on R, then (gof) (2) + (fog) (–2) is equal to
-10
10
8
-8
The Correct Option is D
Solution and Explanation
\(f(x) = \begin{cases} x + a, & x \leq 0 \\ |x - 4|, & x > 0 \end{cases}\) and\(g(x) = \begin{cases} x + 1, & x < 0 \\ (x - 4)^2 + b, & x \geq 0 \end{cases}\)
∵ f(x) and g(x) are continuous on R
∴ a = 4 and b = 1 – 16 = –15
then (gof)(2) + (fog) (–2)
= g(2) + f(–1)
= –11 + 3 = – 8
So, the correct option is (D): -8
Top Questions on Functions
- Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 \). If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of \( 28 \sum_{i=1}^5 f(i) \) is:
- Let the function $ f(x) = \sqrt{\log_e(1 - x^2)} $. Then the domain of $ f(x) $ is:
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- If the domain of the function $ f(x) = \log_e \left( \frac{2x-3}{5+4x} \right) + \sin^{-1} \left( \frac{4+3x}{2-x} \right) $ is $ [\alpha, \beta] $, then $ \alpha^2 + 4\beta $ is equal to
- Let the domain of the function $ f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2) $ be (a, b). If $ \int_0^{a+b} [x^2] dx = p - q\sqrt{r} $, $ p, q, r \in \mathbb{N} $, gcd(p, q, r) = 1, where [.] is the greatest integer function, then p + q + r is equal to
Questions Asked in JEE Main exam
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): The density of the copper ($^{64}Cu$) nucleus is greater than that of the carbon ($^{12}C$) nucleus.
Reason (R): The nucleus of mass number A has a radius proportional to $A^{1/3}$.
In the light of the above statements, choose the most appropriate answer from the options given below:- JEE Main - 2025
- Nuclear physics
- The roots of the quadratic equation \( 3x^2 - px + q = 0 \) are the 10th and 11th terms of an arithmetic progression with common difference \( \frac{3}{2} \). If the sum of the first 11 terms of this arithmetic progression is 88, then \( q - 2q \) is equal to:
- JEE Main - 2025
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- The type of oxide formed by the element among Li, Na, Be, Mg, B and Al that has the least atomic radius is:
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- The number of real solution(s) of the equation \( x^2 + 3x + 2 = \min \left( |x - 3|, |x + 2| \right) \) is:
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- Coordinate Geometry
- Two iron solid discs of negligible thickness have radii \( R_1 \) and \( R_2 \) and moment of inertia \( I_1 \) and \( I_2 \), respectively. For \( R_2 = 2R_1 \), the ratio of \( I_1 \) and \( I_2 \) would be \( \frac{1}{x} \), where \( x \) is:
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- System of Particles & Rotational Motion
Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions