Advertisements
Advertisements
Question
Check the injectivity and surjectivity of the following function.
f : Z → Z given by f(x) = x2
Solution
f : Z → Z given by f(x) = x2
f(2) = 22 = 4 and f(– 2) = (– 2)2 = 4
∴ f(2) = f(– 2) but 2 ≠ – 2
∴ f is not injective
2 ∈ Z but there is no x ∈ Z such that 2 = f(x) = x2
∴ f is not surjective
∴ f is neither injective nor surjective.
APPEARS IN
RELATED QUESTIONS
Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
- {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}
- {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
- {(1, 3), (1, 5), (2, 5)}
find: f(1), f(−1), f(0) and f(2).
f, g, h are three function defined from R to R as follow:
(iii) h(x) = x2 + 1
Find the range of function.
If \[f\left( x \right) = \frac{1}{1 - x}\] , show that f[f[f(x)]] = x.
If f(x) = loge (1 − x) and g(x) = [x], then determine function:
(ii) fg
If f(x) = loge (1 − x) and g(x) = [x], then determine function:
(iii) \[\frac{f}{g}\]
Write the range of the function f(x) = sin [x], where \[\frac{- \pi}{4} \leq x \leq \frac{\pi}{4}\] .
If f(x) = cos [π2]x + cos [−π2] x, where [x] denotes the greatest integer less than or equal to x, then write the value of f(π).
If f(x) = 4x − x2, x ∈ R, then write the value of f(a + 1) −f(a − 1).
Let f(x) = |x − 1|. Then,
If \[f\left( x \right) = \frac{\sin^4 x + \cos^2 x}{\sin^2 x + \cos^4 x}\] for x ∈ R, then f (2002) =
The domain of definition of the function \[f\left( x \right) = \sqrt{\frac{x - 2}{x + 2}} + \sqrt{\frac{1 - x}{1 + x}}\] is
Check if the following relation is function:
If f(m) = m2 − 3m + 1, find f(−3)
A function f is defined as follows: f(x) = 4x + 5, for −4 ≤ x < 0. Find the values of f(−1), f(−2), f(0), if they exist.
Check if the following relation is a function.
If f(m) = m2 − 3m + 1, find f(0)
Find the domain and range of the follwoing function.
h(x) = `sqrt(x + 5)/(5 + x)`
Find the domain and range of the following function.
f(x) = `sqrt((x - 2)(5 - x)`
Express the area A of circle as a function of its circumference C.
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 : R → R given by f(x) = x3
Express the following exponential equation in logarithmic form
25 = 32
Find the domain of f(x) = ln (x − 5)
Write the following expression as sum or difference of logarithm
`log (sqrt(x) root(3)(y))`
Answer the following:
A function f : R → R defined by f(x) = `(3x)/5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f–1
Answer the following:
Simplify `log_10 28/45 - log_10 35/324 + log_10 325/432 - log_10 13/15`
Given the function f: x → x2 – 5x + 6, evaluate f(– 1)
A graph representing the function f(x) is given in it is clear that f(9) = 2
Describe the following Domain
A function f is defined by f(x) = 2x – 3 find x such that f(x) = x
The domain of the function f(x) = `sqrtx` is ______.
If f(x) = 5x - 3, then f-1(x) is ______
Domain of function f(x) = cos–1 6x is ______.
Find the domain for which the functions f(x) = 2x2 – 1 and g(x) = 1 – 3x are equal.
Find the domain of the following function.
f(x) = `x/(x^2 + 3x + 2)`
Find the range of the following functions given by `|x - 4|/(x - 4)`
Find the domain of the following functions given by f(x) = `1/sqrt(x + |x|)`
The domain and range of the function f given by f(x) = 2 – |x – 5| is ______.
Let f(θ) = sin θ (sin θ + sin 3θ) then ______.
