Determine whether the function fx2xx3 is of order 2x That is

Determine whether the function, f(x)=2^x+x^3 is of order 2^x.
That is, determine if f(x) is (2^x). [Hint: Determine if f(x) is both O(2^x) and (2^x ).]
Show your work.

Solution

f(x)=2^x+x^3

first to do thislet\'s find out 2^x or x^3 is bigger value when \'x\' is too large

plug x=10 then 2^10 =1024

when x=10 then x^3 =1000

here(1024>1000)

let us plug some more bigger value

plug x=20 then 2^20=1048576

x=20 then x^20 =40400

here also(1048576 >> 40400)

so we can say 2^x is bigger than x^3

now 2^x + x^3 < c2^x ( here c is some random value which satisfy this)

so f(x) = O(2^x)

yes the given function is order of 2^x

to determine if f(x) is (2^x)

we can say this only when c1(2^x) < f(x) < c2(2^x)

here c1 and c2 are any values

so now if we choose some small integer vlaue for c1 and big value for c2

this satisfy

so f(x) = (2^x)