Advertisements
Advertisements
Question
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
Advertisements
Solution
\[\text{We have}, \]
\[f\left( x \right) = \sqrt{25 - x^2}\]
\[\text{The function is defined only when } 25 - x^2 \geq 0\]
\[ \Rightarrow x^2 - 25 \leq 0\]
\[ \Rightarrow \left( x + 5 \right)\left( x - 5 \right) \leq 0\]
\[ \Rightarrow x \in \left[ - 5, 5 \right]\]
\[\text{So, the domain of the given function is} \left[ - 5, 5 \right] .\]
APPEARS IN
RELATED QUESTIONS
Show that the function f : R* → R* defined by f(x) = `1/x` is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R?
Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`
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)}
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 1 + 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}
Let f : N → N be defined by
`f(n) = { (n+ 1, if n is odd),( n-1 , if n is even):}`
Show that f is a bijection.
[CBSE 2012, NCERT]
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
Find fog and gof if : f (x) = x+1, g (x) = sin x .
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2
Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.
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.
Consider the function f : R+ → [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with f -1 (y) = `(sqrt(54 + 5y) -3)/5` [CBSE 2015]
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
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).
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
The function
f : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
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)
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
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.
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
The function f : R → R defined by f(x) = 3 – 4x is ____________.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers 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?
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
Find the domain of sin–1 (x2 – 4).
