Expert: Sherry Wallin - 1/9/2010

Question
I am studying for my Pre-Cal midterm and need help with 3 problems.
For problems A & B I have to: graph the function, find the domain and range, intervals i which its, increasing, decreasing, remaining constant, relative minimum and maximum and determine whether its even, odd, or neither.

A: F(X)= X√X-6

B: G(X)= (X-2) (X≤0)
        (√X-5) (0<X≤2)
        (6) (X>2)

For problem C I have to find the difference quotient.

C: F(X)= X^3+X

Answer
Hi Ellis~
    Make a table of x values and then calculate what y would be for each x and then plot the points and connect the dots. There are other ways using calculus but since you say you are doing precalculus this is the way I would recommend. A is really x^(3/2)-6 so for x = 0 you have y = -6 telling you that (0,-6) is on the graph. Let x = 1 then y = -5 so (1,-5) is on the graph. Let x = 4, so y = 2 so the point (4, 2) is on the graph. You really need to check some values in between the x values I gave you to see what the graph really looks like. As for the domain ask yourself is there any values of x that cannot occur? Hopefully you see that x cannot be negative but there is no other restriction on x so the domain is {x|x>=0 and x is a real number}. Now y depends on x and if x has to be at least 0 or greater that limits what y can be. For example can f(x) = y be -1? Suppose it could be then x^(3/2)-6 = -1-> x^(3/2) = 5, which makes x some where between 1 and 2 so it would be ok. But try y =5 then x^(3/2) - 6 = 5 -> x^(3/2) = 11, which is ok. So is there some y that just cannot be? How about y = -7 -> x^(3/2)- 6 = -7 -> x^(3/2) = -1 and x cannot be negative so the smallest y can be is -6 which means x^(3/2) -6 = -6 -> x^(3/2) = 0 so x = 0 meaning (0, -6) is on the graph. So the range is {y|y>=-6 and y is a real number}. As a favor to you I will tell you that the function is always increasing. Do part B similarly.

For part 3 you need to look at:
[(x+h)^3 +(x+h)-(x^3+x)]/h ->[x^3 +3x^2h +3xh^2 +h^3 +x +h -x^3 -x]/h -> [3x^2h +3xh^2 +h^3 +h]/h
-> 3x^2+3xh +h^2+1

Math Prof