Advertisements
Advertisements
प्रश्न
A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.
Advertisements
उत्तर
We have, f : [– 4, 4] `rightarrow` [0, 4] defined as f(x) = `sqrt(16 - x^2)`
(i) One-One
f(x1) = f(x2)
`\implies sqrt(16 - x_1^2) = sqrt(16 - x_2^2)`
`\implies 16 - x_1^2 = 16 - x_2^2`
`\implies x_1^2 = x_2^2`
`\implies x_1^2 - x_2^2` = 0
`\implies` (x1 + x2) (x1 – x2) = 0
Here, x1 + x2 = 0 is also possible.
As if x1 = 4 and x2 = – 4.
Then, x1 + x2 = 0 is also possible.
∴ x1 = – x2
But for one-one,
x1 = – x2
so, f(x) is not one-one.
(ii) Onto
Let, y = `sqrt(16 - x^2)`
`\implies` y2 = 16 – x2
`\implies` x2 = 16 – y2
`\implies` x = `sqrt(16 - y^2) ∈ [0, 4]`
So, f(x) is onto.
For f(a) = `sqrt(7)`, we have
f(a) = `sqrt(16 - a^2)`
`\implies sqrt(7) = sqrt(16 - a^2)`
On equating both sides
`\implies` 7 = 16 – a2
`\implies` a2 =16 – 7
`\implies` a2 = 9
`\implies` a = ± 3
Hence, possible values of a are 3 and – 3.
संबंधित प्रश्न
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(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)?
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
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.
Which of the following graphs represents a one-one function?
If f : R → R is given by f(x) = x3, write f−1 (1).
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
\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
Mark the correct alternative in the following question:
Let A = {1, 2, ... , n} and B = {a, b}. Then the number of subjections from A into B is
Which function is used to check whether a character is alphanumeric or not?
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R 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?
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- The function f: R → R defined by f(x) = x − 4 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: R → R be defined by f(x) = x2 is:
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 ______.
`x^(log_5x) > 5` implies ______.
Let A = {1, 2, 3, ..., 10} and f : A `rightarrow` A be defined as
f(k) = `{{:(k + 1, if k "is odd"),( k, if k "is even"):}`.
Then the number of possible functions g : A `rightarrow` A such that gof = f is ______.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
