Advertisements
Advertisements
Question
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
Solution
(fog) (x) = f (g (x)) = f(sin x) = sin2 x + sin x + 1
and (gof) (x) = g (f (x)) = g (x2+ x + 1) = sin ( x2+ x + 1)
So, fog ≠ gof.
APPEARS IN
RELATED QUESTIONS
Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f, g: A → B be functions defined by f(x) = x2 − x, x ∈ A and g(x) = `2|x - 1/2|- 1, x in A`. Are f and g equal?
Justify your answer. (Hint: One may note that two functions f: A → B and g: A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions).
Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
Which of the following functions from A to B are one-one and onto?
f1 = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.
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 .`
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.
Let f : R → R and g : R → R be defined by f(x) = x + 1 and g (x) = x − 1. Show that fog = gof = IR.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
If f : R → R is defined by f(x) = x2, find f−1 (−25).
If f : R → R, g : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
A function f from the set of natural numbers to integers defined by
`{([n-1]/2," when n is odd" is ),(-n/2,when n is even ) :}`
Which of the following functions from
to itself are bijections?
Let
\[f : R - \left\{ n \right\} \to R\]
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
If \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
Mark the correct alternative in the following question:
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Which of the following functions from Z into Z is bijective?
Let g(x) = x2 – 4x – 5, then ____________.
The function f : R → R given by f(x) = x3 – 1 is ____________.
Let f: R → R defined by f(x) = x4. Choose the correct answer
If log102 = 0.3010.log103 = 0.4771 then the number of ciphers after decimal before a significant figure comes in `(5/3)^-100` is ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
