The equation of the function is:
f(x)=4-2(x-6)^1/3, for x not equal to 5
calculate precisely the value of f(x).
so I got this:
5=4-2(x-6)^1/3
1=-2(x-6)^1/3
(-1/2)=(x-6)^1/3
(-1/2)^3=(x-6)
-.125=x-6
now the next step is suposed to be this but I don't understand how they got it.
abs[x-5]=0.875
If you solve for x using my orginal simplification I got 5.875 but this istn' right.
You can't solve "precisely" for the value of x unless you say what x is.
You have only said what x is NOT equal to. Is the word NOT there by mistake?
well it said that in the book but it wants you to solve for x IF the function is equal to 5.
<< it wants you to solve for x IF the function is equal to 5. >> That is not the question you wrote at first. That is why I was confused by your question.
If x=5, then
f(5)= 4 -2*(-1)^1/3
The cube root of -1 is -1, so
f(5) = 4+2 = 6
I am suposed to solve for x if the FUNCTION is equal to 5.
To understand how the equation abs[x-5] = 0.875 is obtained, let's go through the steps:
1. From your previous simplification (-0.125 = x - 6), the next step is to isolate the variable x. We can do this by adding 6 to both sides of the equation:
-0.125 + 6 = x - 6 + 6
5.875 = x
2. Now, let's look at the given equation f(x) = 4 - 2(x - 6)^(1/3) for x not equal to 5. When x is equal to 5, we have a different expression. In this case, our expression becomes:
f(5) = 4 - 2(5 - 6)^(1/3)
Notice that (5 - 6)^(1/3) equals -1, as the expression inside the parentheses is negative and raised to an odd root.
3. So, let's substitute this value into the equation:
f(5) = 4 - 2(-1)^(1/3)
Now, the expression (-1)^(1/3) represents the cube root of -1. The cube root of -1 can be written as one of the complex cube roots of unity, which are defined as:
(-1)^(1/3) = cos(2π/3) + i sin(2π/3) ≈ -0.5 + i√(3)/2
So, replacing (-1)^(1/3) with -0.5 + i√(3)/2, we get:
f(5) = 4 - 2(-0.5 + i√(3)/2)
f(5) = 4 + 1 - i√(3)
f(5) = 5 - i√(3)
Therefore, the precise value of f(x) for x = 5 is 5 - i√(3).