Question

Why is f not a function from R to R?

1) f(x) = 1/x
2) f(x) = √x
3) f(x) = ±(x² - 1)

Answer 1

The given expressions are not considered functions from R to R because they either do not produce a real number output for every real number input, or they associate multiple outputs with a single input, violating the definition of a function.

Step-by-step explanation:

The question asks why f is not a function from R to R for three given expressions. In mathematics, for a rule to be considered a function from the set of real numbers (R) to itself, every input must be associated with exactly one output.

• For f(x) = 1/x, the rule breaks down when x equals zero since division by zero is undefined. There is no real number that can be the output when x is zero, so f is not a function from R to R.

• The square root f(x) = √x is not defined for negative numbers within the real numbers. This means the output is not a real number for any input x which is negative, hence f is not a function from R to R.

• f(x) = ±(x² - 1) specifies two possible outputs for each input (the positive and negative square roots), making f not a function because a single input should not be mapped to multiple outputs.