Which expression is defined for all real numbers?
a) x + 3/x^2
b) 5/(x + 2)^2
c)4/(x − 2)^2
d)7/x^2 + 3
any time there is a fraction, the function is undefined when the denominator is zero.
None of the choices is defined for all real numbers.
The expression defined for all real numbers is:
d) 7/x^2 + 3
This expression is defined for all real numbers because it does not contain any variables in the denominator and the only restriction is that x should not be equal to 0.
To determine which expression is defined for all real numbers, we need to identify any values of x that would make each expression undefined.
a) The expression x + 3/x^2 is undefined when x = 0 because division by zero is not defined. Therefore, this expression is not defined for all real numbers.
b) The expression 5/(x + 2)^2 is defined for all real numbers except when x = -2 because division by zero is not defined. Therefore, this expression is defined for all real numbers except -2.
c) The expression 4/(x - 2)^2 is defined for all real numbers except when x = 2 because division by zero is not defined. Therefore, this expression is defined for all real numbers except 2.
d) The expression 7/x^2 + 3 is undefined when x = 0 because division by zero is not defined. Therefore, this expression is not defined for all real numbers.
Based on the analysis above, the expression that is defined for all real numbers is option b) 5/(x + 2)^2, except when x = -2.