Advertisements
Advertisements
Question
Check the injectivity and surjectivity of the following function.
f : N → N given by f(x) = x3
Solution
f : N → N given by f(x) = x3
Let f(x1) = f(x2)
∴ x13 = x23
∴ x13 – x23 = 0
∴ `(x_1 - x_2) underbrace((x_1^2 + x_1 x_2 + x_2^2))_(> 0 "for all" "x"_1, "x"_2 "as it's discriminant" < 0)` = 0
∴ x1 = x2
∴ f is injective.
Numbers from codomain which are not cubes of natural numbers are not images under f.
∴ f is not surjective.
APPEARS IN
RELATED QUESTIONS
Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine
(b) {x : f(x) = −2}
If \[f\left( x \right) = \frac{1}{1 - x}\] , show that f[f[f(x)]] = x.
If \[f\left( x \right) = \begin{cases}x^2 , & \text{ when } x < 0 \\ x, & \text{ when } 0 \leq x < 1 \\ \frac{1}{x}, & \text{ when } x \geq 1\end{cases}\]
find: (a) f(1/2), (b) f(−2), (c) f(1), (d)
Write the domain and range of function f(x) given by \[f\left( x \right) = \sqrt{\left[ x \right] - x}\] .
Which one of the following is not a function?
If A = {1, 2, 3} and B = {x, y}, then the number of functions that can be defined from A into B is
If f(x) = cos (log x), then value of \[f\left( x \right) f\left( 4 \right) - \frac{1}{2} \left\{ f\left( \frac{x}{4} \right) + f\left( 4x \right) \right\}\] is
If x ≠ 1 and \[f\left( x \right) = \frac{x + 1}{x - 1}\] is a real function, then f(f(f(2))) is
If f : R → R be given by for all \[f\left( x \right) = \frac{4^x}{4^x + 2}\] x ∈ R, then
The range of the function \[f\left( x \right) = \frac{x}{\left| x \right|}\] is
Let \[f\left( x \right) = \sqrt{x^2 + 1}\ ] . Then, which of the following is correct?
A function f is defined as follows: f(x) = 5 − x for 0 ≤ x ≤ 4. Find the value of x such that f(x) = 3
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.
{(1, 0), (3, 3), (2, −1), (4, 1), (2, 2)}
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.
{(1, 1), (2, 1), (3, 1), (4, 1)}
If f(m) = m2 − 3m + 1, find f(x + 1)
Find x, if f(x) = g(x) where f(x) = `sqrt(x) - 3`, g(x) = 5 – x
Express the area A of a square as a function of its side s
Show that if f : A → B and g : B → C are one-one, then g ° f is also one-one
Express the following logarithmic equation in exponential form
ln 1 = 0
Express the following logarithmic equation in exponential form
In `1/2` = – 0.693
If `log((x + y)/3) = 1/2 log x + 1/2 logy`, show that `x/y + y/x` = 7
Select the correct answer from given alternatives.
Find x, if 2log2 x = 4
The equation logx2 16 + log2x 64 = 3 has,
A function f is defined as : f(x) = 5 – x for 0 ≤ x ≤ 4. Find the value of x such that f(x) = 3
Answer the following:
If f(x) = 3x + a and f(1) = 7 find a and f(4)
Answer the following:
Find x, if x = 33log32
Answer the following:
If `log ((x - y)/5) = 1/2 logx + 1/2 log y`, show that x2 + y2 = 27xy
Answer the following:
Find (f ° g) (x) and (g ° f) (x)
f(x) = `x/(x + 1)`, g(x) = `x/(1 - x)`
The range of 7, 11, 16, 27, 31, 33, 42, 49 is ______.
The domain of the real valued function f(x) = `sqrt((x - 2)/(3 - x))` is ______.
If f(x) = `1/sqrt(4 - 3x)`, then dom(f) = ______..
Find the domain of the following function.
f(x) = `x/(x^2 + 3x + 2)`
Find the domain of the following functions given by f(x) = x|x|
Domain of `sqrt(a^2 - x^2) (a > 0)` is ______.
The domain and range of the real function f defined by f(x) = `(4 - x)/(x - 4)` is given by ______.
The function f: R `rightarrow` R defined by f(x) = sin x is ______.
Which of the following functions is NOT one-one?
The range of the function f(x) = `""^(7 - x)P_(x - 3)` is ______.
Range of the function f(x) = `x/(1 + x^2)` is ______.
