Advertisements
Advertisements
Question
If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))
Advertisements
Solution
Given that f(x)= x2 – 3x + 2
f(f(x) = f(x2 – 3x + 2)
= (x2 – 3x + 2)2 – 3(x2 – 3x + 2) + 2
= x4 + 9x2 + 4 – 6x3 – 12x + 4x2 – 3x2 + 9x – 6 + 2
= x4 + 10x2 – 6x3 – 3x
f(f(x)) = x4 – 6x3 + 10x2 – 3x
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : N → N 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 : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 3), (b, 2), (c, 1)}
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Which of the following functions from A to B are one-one and onto?
f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|
Set of ordered pair of a function? If so, examine whether the mapping is injective or surjective :{(x, y) : x is a person, y is the mother of x}
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)?
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
Let \[f : \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \to R\] be a function defined by f(x) = cos [x]. Write range (f).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)} [NCERT EXEMPLAR]
Let\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = \text{B and C} = \left\{ x \in R : x \geq 0 \right\} and\]\[S = \left\{ \left( x, y \right) \in A \times B : x^2 + y^2 = 1 \right\} \text{and } S_0 = \left\{ \left( x, y \right) \in A \times C : x^2 + y^2 = 1 \right\}\]
Then,
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is
Which of the following functions from
to itself are bijections?
Let
\[f : R - \left\{ n \right\} \to R\]
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
Which function is used to check whether a character is alphanumeric or not?
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
The smallest integer function f(x) = [x] is ____________.
Let f : R → R, g : R → R be two functions such that f(x) = 2x – 3, g(x) = x3 + 5. The function (fog)-1 (x) is equal to ____________.
Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R 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 f: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
`x^(log_5x) > 5` implies ______.
If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) C ∩ Z is ______.
A function is called bijective if it is both:
