which functions have a domain of all real numbers?

Question

A. f(x)=2(x+2)(x-1) B.f(x)=(x-1)2 -2 C.f(x)= (x-1)(x+3) D. f(x)=-(x+3)2 E. f(x)=-2x2-8x+3 F. f(x)=1/2x2-2

Answer 1

A - E has a domain of all real numbers.

 

F is a rational function.  Its denominator term is (2x² - 2).  Therefore, the domain is found by setting this term equal to zero.

 

2x² - 2 = 0

 

2(x² - 1) = 0

 

2(x - 1)(x + 1) = 0

 

x = -1     and     x = 1

 

 

The domain for this function is all real numbers except for x=-1 and x=1.

 

 

Or if you interpret the function as     f(x) = (1/2x²) - 2

 

then the domain is real numbers except for x=0.

Answer 2

All but F

Answer 3

Domain refers to the set of values x can assume in each of these functions. Clearly x can assume all real values in each of the functions shown above.