Let \(ƒ : R→R\) be defined as \(ƒ(x) = x^3 + x – 5\) . If g(x) is a function such that \(ƒ(g(x)) = x\), ∀‘x‘∈R. Then \(g^′(63)\) is equal to_____.
\(\frac {1}{49}\)
\(\frac {3}{49}\)
\(\frac {43}{49}\)
\(\frac {91}{49}\)
The Correct Option is A
Solution and Explanation
\(ƒ(x) = 3x^2 + 1\)
\(ƒ^′(x)\) is bijective function
and \(ƒ(g(x)) = x⇒g(x\)) is inverse of \(ƒ(x)\)
\(g(ƒ(x)) = x\)
\(g^′(f(x)).f^′(x) = 1\)
\(g^′(f(x)) = \frac {1}{3x^2+1}\)
Put \(x = 4\) we get
\(g^′(63)=\frac {1}{49}\)
So, the correct option is (A): \(\frac {1}{49}\)
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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
