How many real numbers are not in the domain of the function
f(x) = 1/(x - 64) + 1/(x^2 - 64) + 1/(x^3 - 64) + 1/(x^4 - 64)
The real numbers that are not in the domain of the function f(x)=(1x−64)+(1x2−64)+(1x3−64)f(x)=(1x−64)+(1x2−64)+(1x3−64) are the numbers that will make the function undefined.
We look at the fraction denominators. The function will be undefined when either of the denominators is equal to zero.
Case 1:
x−64=0x−64=0
Add 64 to both sides and simplify:
x−64+64=0+64⟹x=64x−64+64=0+64⟹x=64
Case 2:
x2−64=0x2−64+64=0+64x2=64⟹x=±√64∴x=8,x=−8x2−64=0x2−64+64=0+64x2=64⟹x=±64∴x=8,x=−8
Case 3:
x3−64=0x3−64+64=0+64x3=64⟹x=3√64∴x=4x3−64=0x3−64+64=0+64x3=64⟹x=643∴x=4
Hence the real numbers that are not in the domain of the given function are −8,4,8−8,4,8 and 64
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