Advertisements
Advertisements
Question
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
Options
\[\sqrt{2}\]
\[- \sqrt{2}\]
1
-1
Advertisements
Solution
(d) −1
\[f\left( f\left( x \right) \right) = x\]
\[ \Rightarrow f\left( \frac{\alpha x}{x + 1} \right) = x\]
\[ \Rightarrow \frac{\alpha\left( \frac{\alpha x}{x + 1} \right)}{\left( \frac{\alpha x}{x + 1} \right) + 1} = x\]
\[ \Rightarrow \frac{\alpha^2 x}{\alpha x + x + 1} = x\]
\[ \Rightarrow \alpha^2 x = \alpha x^2 + x^2 + x\]
\[ \Rightarrow \alpha^2 x - \alpha x^2 - x^2 - x = 0\]
\[ \Rightarrow \alpha^2 x - \alpha x^2 - \left( x^2 + x \right) = 0\]
\[\text{Solving } \text{for } \text{ the } \alpha \text{ we get}, \]
\[\alpha = \frac{- \left( - x^2 \right) \pm \sqrt{\left( - x^2 \right)^2 - 4 \times x \times \left[ - \left( x^2 + x \right) \right]}}{2x}\]
\[ = \frac{x^2 \pm \sqrt{x^4 + 4 x^3 + 4 x^2}}{2x}\]
\[ = x + 1, - 1\]
\[\text{Here, - 1 is independent of } x, \]
\[ \therefore for, \alpha = - 1, f\left( f\left( x \right) \right) = x\]
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x3
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
Show that the exponential function f : R → R, given by f(x) = ex, is one-one but not onto. What happens if the co-domain is replaced by`R0^+` (set of all positive real numbers)?
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R is given by (f_1/f_2) (x) = (f_1(x))/(f_2 (x)) for all x in R .`
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
If f : R → R is defined by f(x) = x2, write f−1 (25)
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If the function
\[f : R \to R\] be such that
\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Write about strlen() function.
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
k = {(1,4), (2, 5)}
Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
f(x) = `x/2`
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
The smallest integer function f(x) = [x] is ____________.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let f: R → R be defined by f(x) = x2 is:
A function f: x → y is said to be one – one (or injective) if:
If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n" "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.
In the function from \(X\) to \(Y\), if \(6\in Y\) is not the image of any element of \(X\), the function is:
Which condition represents an onto (surjective) function?
