Advertisements
Advertisements
Question
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
Advertisements
Solution
` f (x) = ([x - 1])/(x - 1) = (+(x-1)) /(x-1)` = ±1
Range of f = {-1 ,1 }
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
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
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}
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 : Z → Z, defined by f(x) = x − 5
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4
Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.
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.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 8 and g(x) = 3x3 + 1 .
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 .
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.
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
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 = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.
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 : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
If f : R → R is defined by f(x) = x2, find f−1 (−25).
Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by \[f\left( x \right) = \frac{\left| x \right|}{x}\]Write the range of f.
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]
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] 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, 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 A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
k(x) = x2
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Which of the following functions from Z into Z are bijections?
Which of the following functions from Z into Z is bijective?
Let g(x) = x2 – 4x – 5, then ____________.
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers is ____________.
If N be the set of all-natural numbers, consider f: N → N such that f(x) = 2x, ∀ x ∈ N, then f 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: N → N be defined by f(x) = x2 is ____________.
If `f : R -> R^+ U {0}` be defined by `f(x) = x^2, x ∈ R`. The mapping is
A function f: x → y is said to be one – one (or injective) if:
Number of integral values of x satisfying the inequality `(3/4)^(6x + 10 - x^2) < 27/64` is ______.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
