Advertisements
Advertisements
Question
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
Advertisements
Solution
Here f(x) = |x| + x which can be redefined as
f(x) = `{(2x, "if" x ≥ 0),(0, "if" x < 0):}`
Similarly, the function g defined by g(x) = |x| – x may be redefined as
g(x) = `{(0, "if" x ≥ 0),(-2x, "if" x < 0):}`
Therefore, g o f gets defined as:
For x ≥ 0, (g o f) (x) = g (f(x) = g (2x) = 0
and for x < 0, (g o f) (x) = g (f(x) = g (0) = 0.
Consequently, we have (g o f) (x) = 0, ∀ x ∈ R.
Similarly, f o g gets defined as:
For x ≥ 0, (f o g) (x) = f (g(x) = f(0) = 0,
and for x < 0, (f o g) (x) = f (g(x)) = f(–2x) = – 4x.
i.e. (f o g) (x) = `{(0, x > 0),(-4x, x < 0):}`
APPEARS IN
RELATED QUESTIONS
Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f-1
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
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 ∈ 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.)
Give an example of a function which is neither one-one nor onto ?
Classify the following function as injection, surjection or bijection :
f : Q → Q, defined by f(x) = x3 + 1
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
Find fog and gof if : f (x) = x+1, g (x) = sin x .
Let f be a real function given by f (x)=`sqrt (x-2)`
Find each of the following:
(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2
Also, show that fof ≠ `f^2` .
If f : R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1. Hence, find f−1(24) and f−1 (5).
If f : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.
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 : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Write the domain of the real function
`f (x) = sqrtx - [x] .`
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
The function
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
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
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
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
f = {(1, 4), (1, 5), (2, 4), (3, 5)}
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
g = {(1, 4), (2, 4), (3, 4)}
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- Mr. ’X’ and his wife ‘W’ both exercised their voting right in the general election-2019, Which of the following is true?
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: N → N be defined by f(x) = x2 is ____________.
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 : N → R be defined by f(x) = x2. Range of the function among the following is ____________.
If f; R → R f(x) = 10x + 3 then f–1(x) is:
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
If f: R→R is a function defined by f(x) = `[x - 1]cos((2x - 1)/2)π`, where [ ] denotes the greatest integer function, then f is ______.
Let f: R→R be a continuous function such that f(x) + f(x + 1) = 2, for all x ∈ R. If I1 = `int_0^8f(x)dx` and I2 = `int_(-1)^3f(x)dx`, then the value of I1 + 2I2 is equal to ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
A function is called many-one if two or more different elements of the domain have:
A function is onto when every element of the codomain has:
