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Show that the exponential function `f: R->R` , given by `f(x)=e^x` , is one-one but not onto. What happens if the co-domain is replaced by `R^+` (set of all positive real numbers).

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`f:R→R`, given by `f(x)=e ^x` . one−one: Let `x _1` ​ and `x ...
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RD SHARMA-FUNCTION-Solved Examples And Exercises
  1. Let A={1,\ 2,\ 3} . Write all one-one from A to itself.

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  2. If f: R->R be the function defined by f(x)=4x^3+7 , show that f is a b...

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  3. Show that the exponential function f: R->R , given by f(x)=e^x , is ...

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  4. Show that the logarithmic function f: R0+->R given by f(x)=(log)a x...

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  5. Show that a one-one function f : {1, 2, 3}-> {1, 2, 3}must be onto.

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  6. If A={1,\ 2,\ 3} , show that an onto function f: A->A must be one-o...

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  7. Find the number of all onto functions from the set A={1,\ 2,\ 3,\ ...

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  8. Give examples of two one-one functions f1 and f2 from R to R such t...

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  9. Give examples of two surjective function f1 and f2 from Z to Z such...

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  10. Show that if f1 and f2 are one-one maps from R to R , then the product...

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  11. Suppose f1 and f2 are non-zero one-one functions from R to R . Is (...

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  12. Given A={2,\ 3,\ 4} , B={2,\ 5,\ 6,\ 7} . Construct an example of a...

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  13. Given A={2,\ 3,\ 4} , B={2,\ 5,\ 6,\ 7} . Construct an example of a...

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  14. Given A={2,\ 3,\ 4} , B={2,\ 5,\ 6,\ 7} . Construct an example of a...

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  15. Show that f: R->R , given by f(x)=x-[x] , is neither one-one nor onto.

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  16. Let f: N->N be defined by: f(n)={n+1,\ if\ n\ i s\ od d n-1,\ if\ n\ i...

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  17. Let R be the set of real numbers. If f: R->R :f(x)=x^2 and g: R->R ...

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  18. Let : R->R ; f(x)=sinx and g: R->R ; g(x)=x^2 find fog and gof .

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  19. Let f:{2,3,4,5}vec{3,4,5,9}a n dg:{3,4,5,9}vec{7, 11 , 15} be function...

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  20. Let f:{1,\ 3,4\ }->{1,\ 2,\ 5} and g:{1,\ 2,\ 5}->{1,\ 3} be given by ...

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