Home
Class 12
MATHS
For real x, let f(x)""=""x^3+""5x""+""1 ...

For real x, let `f(x)""=""x^3+""5x""+""1` , then (1) f is oneone but not onto R (2) f is onto R but not oneone (3) f is oneone and onto R (4) f is neither oneone nor onto R

A

`f` is one-one but not onto R

B

`f` is onto R but not one-one

C

`f` is one-one and onto R

D

`f` is neither one-one nor onto R

Text Solution

Verified by Experts

The correct Answer is:
C
Given `f(x)=x^(3)+5x+1`
Now `f'(x) =3x^(2)+5 gt 0 AA x in R`
Therefore, `f(x)` is a strictly increasing function, and so it is one-one.
Clearly, `f(x)` is a continuous function and also increasing on R.
`underset (x to -oo)(lim)f(x)= -oo and underset(xto oo)(lim)f(x)=oo`
Hence, `f(x)` takes every volue between `-oo and oo .`
Thus, `f(x)` is an onto function.
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    CENGAGE|Exercise Exercise (Numerical)|31 Videos

  • Quadratic Equations, Inequalities, Modulus and Logarithms

    CENGAGE|Exercise Question Bank|28 Videos

  • SCALER TRIPLE PRODUCTS

    CENGAGE|Exercise DPP 2.3|11 Videos

Similar Questions

Explore conceptually related problems

The function f:[0,3]->[1,29] , defined by f(x)=2x^3−15x^2+36x+1 , is (a) one-one and onto (b) onto but not one-one (c) one-one but not onto (d) neither one-one nor onto

Show that the function f: R toR. defined as f (x) =x ^(2), is neither one-one nor onto.

Let A be a non- empty set of real numbers and f:A->A be such that f(f(x))=x AA x in R then f(x) is (A) bijection (B) one-one but not onto (C) onto but not one-one (D) neither one-one nor onto

Prove that the function f: R to R, given by f (x) =2x, is one-one and onto.

Let the function f: RrarrR be defined by f(x)=2x+sinx for x in Rdot Then f is (a)one-to-one and onto (b)one-to-one but not onto (c)onto but not-one-to-one (d)neither one-to-one nor onto

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

f(x)={(x", if x is rational"),(0", if x is irrational"):}, g(x)={(0", if x is rational"),(x", if x is irrational"):} Then, (f-g) is (A) one-one and onto (B) one-one but nor onto (C) onto but not one-one (D) neither one-one nor onto

Let f: RrarrRa n dg: RrarrR be two one-one and onto function such that they are the mirror images of each other about the line y=a . If h(x)=f(x)+g(x),t h e nh(x) is (a)one-one and onto (b)only one-one and not onto (c)only onto but not one-one (d)neither one-one nor onto

f: NrarrN, where f(x)=x-(-1)^x . Then f is: (a)one-one and into (b)many-one and into (c)one-one and onto (d)many-one and onto

Are f and g both necessarily onto, if gof is onto ?