Determine where the big O(g(x)) for following functions
• F(x)=2xlogx+1
• F(x)=3x -2x +1
• F(x)=4x -2x -3x +4
• F(x)=( x +1)/( x +4)
• F(x)=(x+1)/(x-4
To determine the big O(g(x)) for the given functions, we need to simplify the functions and then identify the growth rate of each function.
1. F(x) = 2xlog(x) + 1
To simplify this function, we can focus on the highest power of x, which is xlog(x). The highest power determines the growth rate of the function. In this case, the growth rate is O(xlog(x)).
2. F(x) = 3x - 2x + 1
Simplifying this function, we can see that the highest power of x is x. Therefore, the growth rate is O(x).
3. F(x) = 4x - 2x - 3x + 4
By simplifying, we find that the highest power of x is x. Thus, the growth rate is O(x).
4. F(x) = (x + 1)/(x + 4)
In this function, as x approaches infinity, both the numerator and denominator grow at the same rate. The highest power of x in both is 1. Therefore, the growth rate is O(1).
5. F(x) = (x + 1)/(x - 4)
Similarly, for this function, as x approaches infinity, both the numerator and denominator grow equally. The highest power of x in both is 1. Thus, the growth rate is O(1).
Summarizing the results:
1. F(x) = 2xlog(x) + 1 has a growth rate of O(xlog(x)).
2. F(x) = 3x - 2x + 1 has a growth rate of O(x).
3. F(x) = 4x - 2x - 3x + 4 has a growth rate of O(x).
4. F(x) = (x + 1)/(x + 4) has a growth rate of O(1).
5. F(x) = (x + 1)/(x - 4) has a growth rate of O(1).