Advertisements
Advertisements
Question
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
Solution
For x1 , x2 ∈ R, consider
f(x1) = f(x2)
⇒ `x_1/(x_1^2 + 1) = x_2/(x_2^2 + 1)`
⇒ `x_1 x_2^2 + x_1 = x_2 x_1^2 + x_2`
⇒ x1 x2 (x2 – x1) = x2 – x1
⇒ x1 = x2 or x1 x2 = 1
We note that there are point, x1 and x2 with x1 ≠ x2 and if f(x1) = f(x2), for instance, If we take x1 = 2 and x2 = `1/2`, then we have f(x1) = `2/5` and f(x2) = `2/5` but `2 ≠ 1/2`.
Hence f is not one-one. Also, f is not onto for if so then for 1∈R ∃ x ∈ R such that f(x) = 1 which gives `x/(x^2 + 1)` = 1
But there is no such x in the domain R, since the equation x2 – x + 1 = 0 does not give any real value of x.
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f: N → N given by f(x) = x2
Following the case, state whether the function is one-one, onto, or bijective. Justify your answer.
f: R → R defined by f(x) = 1 + x2
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 f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
Classify the following function as injection, surjection or bijection : f : N → N 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)?
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
If f(x) = |x|, prove that fof = f.
Let A = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x2 and g(x) = sin (π x/2). Show that g−1 exists but f−1 does not exist. Also, find g−1.
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : R → R is defined by f(x) = x2, write f−1 (25)
If f : R → R is defined by f(x) = x2, find f−1 (−25).
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is
A function f from the set of natural numbers to integers defined by
`{([n-1]/2," when n is odd" is ),(-n/2,when n is even ) :}`
If the function\[f : R \to \text{A given by} f\left( x \right) = \frac{x^2}{x^2 + 1}\] is a surjection, then A =
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 \[f \left( f\left( x \right) \right) = x?\]
Write about strcmp() function.
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 ______.
The smallest integer function f(x) = [x] is ____________.
Which of the following functions from Z into Z is bijective?
Given a function If as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R then 'f' is
If f: R→R is a function defined by f(x) = `[x - 1]cos((2x - 1)/2)π`, where [ ] denotes the greatest integer function, then f is ______.
Consider a set containing function A= {cos–1cosx, sin(sin–1x), sinx((sinx)2 – 1), etan{x}, `e^(|cosx| + |sinx|)`, sin(tan(cosx)), sin(tanx)}. B, C, D, are subsets of A, such that B contains periodic functions, C contains even functions, D contains odd functions then the value of n(B ∩ C) + n(B ∩ D) is ______ where {.} denotes the fractional part of functions)
Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals ______.
