Advertisements
Advertisements
प्रश्न
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
Advertisements
उत्तर
f : R → R is defined by f(x) = 3 – 4x
Let x1, x2 ∈ R such that f(x1) = f(x2)
⇒ 3 – 4x1 = 3 – 4x2
⇒ –4x1 = –4x2
⇒ x1 = x2
∴ f is one-one.
f : R → R be given for every y ∈ R (co-domain of f), there exists an element x ∈ R (domain of f) such that
f(x) = y
⇒ y = 3 – 4x
For any real number y ∈ R, there exists `x = (3 - y)/4 ∈ R` such that:
∴ `f((3 - y)/4)`
= `3 - 4 ((3 - y)/4)`
= 3 – 3 + y
= y
∴ f is onto.
Thus, f is one-one and onto and hence bijective.
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.
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}
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Classify the following function as injection, surjection or bijection :
f : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.
Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
Find fog and gof if : f (x) = x2 g(x) = cos x .
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
If f : R → R defined by f(x) = 3x − 4 is invertible, then write f−1 (x).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
Let the function
\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]
\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Let
\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
If the function
\[f : R \to R\] be such that
\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is
Write about strlen() function.
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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
Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.
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
k = {(1,4), (2, 5)}
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
The function f : R → R defined by f(x) = 3 – 4x 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 ____________.
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 ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
If `f : R -> R^+ U {0}` be defined by `f(x) = x^2, x ∈ R`. The mapping is
Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then
Assertion (A): f(x) has a minimum at x = 1.
Reason (R): When `d/dx (f(x)) < 0, ∀ x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀ x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.
Find the domain of sin–1 (x2 – 4).
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
Many students choosing the same favourite subject resembles which type of function?
Which condition represents a many-one function?
