Advertisements
Advertisements
Question
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
Options
one-one and onto
many-one and onto
one-one and into
many-one and into
Advertisements
Solution
(d) many-one and into
Graph for the given function is as follows.
A line parallel to X axis is cutting the graph at two different values.
Therefore, for two different values of x we are getting the same value of y.
That means it is many one function.[2 , ∞ ) and R is the co-domain of the given function.
Hence, Co-domain ≠ Range Therefore, the given function is into.
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x3
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x3
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is a bijective function.
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
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.)
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 : Z → Z, defined by f(x) = x2 + x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 − x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 3 − 4x
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.
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
Find fog and gof if : f (x) = x+1, g (x) = sin x .
Find fog and gof if : f(x) = `x^2` + 2 , g (x) = 1 − `1/ (1-x)`.
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−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 A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.
If f : R → R is given by f(x) = x3, write f−1 (1).
If f : C → C is defined by f(x) = x4, write f−1 (1).
If f : R → R is defined by f(x) = x2, find f−1 (−25).
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Write the domain of the real function
`f (x) = sqrt([x] - x) .`
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
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 the mapping f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}, given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, then write fog. [NCERT EXEMPLAR]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Let
\[f : R - \left\{ n \right\} \to R\]
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
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.
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
f(x) = `x/2`
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
Which of the following functions from Z into Z is bijective?
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.
- The function f: Z → Z defined by f(x) = x2 is ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
