Advertisements
Advertisements
Question
Let
Options
one-one but not onto
one-one and onto
onto but not one-one
neither one-one nor onto
Advertisements
Solution
Injectivity:
Let x and y be two elements in the domain (R), such that
\[f\left( x \right) = f\left( y \right)\]
\[ \Rightarrow \frac{x^2 - 8}{x^2 + 2} = \frac{y^2 - 8}{y^2 + 2}\]
\[ \Rightarrow \left( x^2 - 8 \right)\left( y^2 + 2 \right) = \left( x^2 + 2 \right)\left( y^2 - 8 \right)\]
\[ \Rightarrow x^2 y^2 + 2 x^2 - 8 y^2 - 16 = x^2 y^2 - 8 x^2 + 2 y^2 - 16\]
\[ \Rightarrow 10 x^2 = 10 y^2 \]
\[ \Rightarrow x^2 = y^2 \]
\[ \Rightarrow x = \pm y \]
So, f is not one-one .
Surjectivity:
\[ \text{ and} f\left( 1 \right) = \frac{\left( 1 \right)^2 - 8}{\left( 1 \right)^2 + 2} = \frac{1 - 8}{1 + 2} = \frac{- 7}{3}\]
The correct answer is (d).
APPEARS IN
RELATED QUESTIONS
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, 2), (b, 1), (c, 1)}
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
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 f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
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)}.
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) = x2, find f−1 (−25).
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
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 ) :}`
If the function\[f : R \to \text{A given by} f\left( x \right) = \frac{x^2}{x^2 + 1}\] is a surjection, then A =
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
The function
The inverse of the function
\[f : R \to \left\{ x \in R : x < 1 \right\}\] given by
\[f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] is
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
If \[g\left( x \right) = x^2 + x - 2\text{ and} \frac{1}{2} gof\left( x \right) = 2 x^2 - 5x + 2\] is equal to
Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by \[f\left( x \right) = \frac{x - 2}{x - 3}, \forall x \in A\] Show that f is bijective. Also, find
(i) x, if f−1(x) = 4
(ii) f−1(7)
Write about strlen() function.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}
Which of the following functions from Z into Z is bijective?
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers is ____________.
The function f: R → R defined as f(x) = x3 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 know among those relations, how many functions can be formed from B to G?
Let x is a real number such that are functions involved are well defined then the value of `lim_(t→0)[max{(sin^-1 x/3 + cos^-1 x/3)^2, min(x^2 + 4x + 7)}]((sin^-1t)/t)` where [.] is greatest integer function and all other brackets are usual brackets.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
Let f(1, 3) `rightarrow` R be a function defined by f(x) = `(x[x])/(1 + x^2)`, where [x] denotes the greatest integer ≤ x, Then the range of f is ______.
Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to ______.
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
