Home
Class 12
MATHS
If f:[0,oo) to [0,1), " and " f(x)=(x)/...

If ` f:[0,oo) to [0,1), " and " f(x)=(x)/(1+x)` then check the nature of the function.

Text Solution

Verified by Experts

Given that `f:[0,oo) to [0, oo),f(x)=(x)/(x+1)`
Let `f(x_(1))=f(x_(2))`
`implies (x_(1))/(x_(1)+1)=(x_(2))/(x_(2)+1)`
`implies x_(1)x_(2)+x_(1)=x_(1)x_(2)+x_(2)`
`implies x_(1)=x_(2)`.
Thus f(x) is one-one.
Now let `y=(x)/(1+x)`
`implies y+yx=x`
`implies x=(y)/(1-y)`
As ` x ge 0, (y)/(1-y) ge 0`
`implies (y)/(1-y) le 0`
`implies 0 le y lt 1` or range of f(x) is `[0,1).`
Thus f(x) is onto.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE PUBLICATION|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

If f:R to R be defined by f(x) =2x+sinx for x in R , then check the nature of the function.

If f:[0,oo)->[0,oo) and f(x)=x/(1+x), then f(x) is (a) one-one and onto (b)one-one but not onto (c)onto but not one-one (d)neither on-one nor onto

Knowledge Check

  • Given f(0)=0 and f(x)=(1)/(1-e^(-(1)/(x))) for x ne 0 . Then the function f(x) is -

    A
    both continuous and differentiable at x=0
    B
    continuous but not differentiable at x=0
    C
    continuous at x=0
    D
    not continuous at x=0

    • Similar Questions

    Explore conceptually related problems

    If f(x) = {(|sinx|/x, when x=0),(1, when x!=0):} then examine the continuity of the function at x=0.

    For x in RR - {0, 1}, let f_1(x) =1/x, f_2(x) = 1-x and f_3(x) = 1/(1-x) be three given functions. If a function, J(x) satisfies (f_2oJ_of_1)(x) = f_3(x) then J(x) is equal to :

    If f(x)={{:((|sinx|)/(x),"when "xne0),(1,"when "x=0):} then examine the continuity of the function at x = 0.

    Consider the function f:(-oo,oo)to(-oo,oo) defined by f(x)=(x^2-a)/(x^2+a),a >0, which of the following is not true?(a) maximum value of f is not attained even though f is bounded. (b) f(x) is increasing on (0,oo) and has minimum at x=0 (c) f(x) is decreasing on (-oo,0) and has minimum at x=0. (d) f(x) is increasing on (-oo,oo) and has neither a local maximum nor a local minimum at x=0.

    Let f(x)={(log(1+x)^(1+x)-x)/(x^2)}dot Then find the value of f(0) so that the function f is continuous at x=0.

    If f(x+y)=f(x)dotf(y) for all real x , ya n df(0)!=0, then prove that the function g(x)=(f(x))/(1+{f(x)}^2) is an even function.

    f:(2,3)->(0,1) defined by f(x)=x-[x] ,where [dot] represents the greatest integer function.

    Home
    Profile