Let
\(f(x)=\frac{x−1}{x+1},x∈R− \left\{0,−1,1\right\}\)
If ƒn+1(x) = ƒ(ƒn(x)) for all n∈N, then ƒ6(6) + ƒ7(7) is equal to :
Let
\(f(x)=\frac{x−1}{x+1},x∈R− \left\{0,−1,1\right\}\)
If ƒn+1(x) = ƒ(ƒn(x)) for all n∈N, then ƒ6(6) + ƒ7(7) is equal to :
\(\frac{7}{6}\)
\(-\frac{3}{2}\)
\(\frac{7}{12}\)
\(-\frac{11}{12}\)
The Correct Option is B
Solution and Explanation
The correct answer is (B) : \(-\frac{3}{2}\)
\(f(x) = \frac{x - 1}{x + 1} \Rightarrow f(f(x)) = \frac{\left(\frac{x - 1}{x + 1}\right) - 1}{\left(\frac{x - 1}{x + 1}\right) + 1} = -\frac{1}{x}\)
\( \Rightarrow f^3(x) = -\frac{x + 1}{x - 1} \Rightarrow f^4(x) = -\frac{\left(\frac{x - 1}{x + 1} + 1\right)}{\left(\frac{x - 1}{x + 1} - 1\right)} = x\)
So, ƒ6(6) + ƒ7(7) = ƒ2(6) + ƒ3(7)
\(= -\frac{1}{6}-\frac{7+1}{7-1}= - \frac{9}{6}=-\frac{3}{2}\)
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Concepts Used:
Composite of Functions
The composite function refers to the resultant value of two specified functions. When the output derived from the application of a function with a second independent variable function becomes the input of the third function, then it is called a composite function. Also, whose scope includes the values of the independent variable for which the result of the first function is placed in the domain of the second.
In Mathematics, the composition of a function is a process, where two functions say f and g create a new function say h in such a way that h (x) = g (f (x)). Here, we can see function g applies to the function of x i.e., f (x)
Let f: A → B and g: B → C are two functions.
So, the composition of f and g, denoted by gof, is known as the function:
g of: A → C given by gof (x) = g (f (x)), A x ∈ A.
Properties of Composite Functions:
- The composition of functions such as f, g, and h is composable because there is some relation or association between functions. The composite function always has associative property. If f º (g º h) = (f º g) º h.
- In special circumstances, some particular functions have commutativity a special property, For example, |x| + 3 = |x + 3| only when x ≥ 0. The functions g and f are said to commute with each other if g º f = f º g.
- The composition of one-to-one (injective) functions is always one-to-one.
- Likewise, onto (surjective) function composition is always onto.
- The inverse composition of the functions f and g are equal to the inverse composition of both functions., like (f º g)−1 = g−1º f−1.