Question

Find the smallest integer n for an O(xn) estimate of order of the function f(x)

f(x) = (x3+ x2 + 5)(x4 - 3x2)/[(2x2 + 2x - 3)(4x2 + 7)];

Answer 1

Answer with explanation:

`f(x)=((x^3+x^2+5)(x^4-3x^2))/((2x^2+2x-3)(4x^2+7))\\\\f(x)=(x^3* (x^4-3x^2)+x^2 * (x^4-3x^2)+5 * (x^4-3x^2))/(2x^2* (4x^2+7)+2x(4x^2+7)-3* (4x^2+7))\\\\f(x)=(x^7-3x^5+x^6-3x^4+5x^4-15x^2)/(8x^4+14x^2+8x^3+14 x-12x^2-21)\\\\f(x)=(x^7+x^6-3x^5+2x^4-15x^2)/(8x^4+8x^3+2x^2+14 x-21)`

The largest degree of numerator is 7 , while the largest degree of denominator is 4.So, as we know

`(x^a)/(x^b)=x^(a-b)\\\\(x^7)/(x^4)=x^(7-4)\\\\=x^3`

→So, order of the function is the highest degree in the function raised to power.Highest degree is 3,when you will reduce the function in Expression form.

So, Degree=3

Order=1