Advertisements
Advertisements
प्रश्न
If f : R → R is defined by f(x) = x2, find f−1 (−25).
Advertisements
उत्तर
\[Let f^{- 1} \left( - 25 \right) = x\]
\[ \Rightarrow f\left( x \right) = - 25\]
\[ \Rightarrow x^2 = - 25\]
\[\text{We cannot find x } \in R,\text{such that} x^2 = - 25 \left( \text{ as x}^2 \geq0 \text{ for all } x \in R \right)\]
\[So, f^{- 1} \left( - 25 \right) = \phi\]
APPEARS IN
संबंधित प्रश्न
Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and f, g : A → B be functions defined by f(x) = x2 – x, x ∈ A and g(x) = `2|x - 1/2| – 1`, x ∈ A. Are f and g equal?
Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions.)
Give an example of a function which is one-one but not onto ?
Give an example of a function which is neither one-one nor onto ?
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
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.
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
Let f : R → R and g : R → R be defined by f(x) = x2 and g(x) = x + 1. Show that fog ≠ gof.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).
If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
Let \[f : \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \to R\] be a function defined by f(x) = cos [x]. Write range (f).
If f : R → R, g : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
Let
f : R → R be given by
\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]
where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
(d) one-one and onto
Which of the following functions form Z to itself are bijections?
The function
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
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 \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
Let
\[f : R \to R\] be given by \[f\left( x \right) = x^2 - 3\] Then, \[f^{- 1}\] is given by
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 X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
g = {(1, 4), (2, 4), (3, 4)}
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
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 ____________.
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.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
If f; R → R f(x) = 10x + 3 then f–1(x) is:
If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n" "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
Let f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then ______.
