Consider the function f:R→R, with the rule f(x). Which of the following would be a valid choice for the rule f(x)?

Question

Consider the function f:R→R, with the rule f(x). Which of the following would be a valid choice for the rule f(x)?

BlackTom AI · Accepted Answer

The question asks: Consider the function f: R → R, with the rule f(x). Which of the following would be a valid choice for the rule f(x)?

First, we need to note that the provided data includes an answer candidate—x^2 + 5—but there are no answer options listed to compare against. Without the set of choices, we cannot definitively select which one is valid versus invalid among multiple options. Nonetheless, we can discuss the criteria for a valid rule f(x) in this context.

Key criteria for a valid f(x):
- Domain compatibility: The expression defining f(x) must be well-defined for every x in the domain, which is R in this case. That means no operations that are undefined for some real x (like taking even roots of negative numbers, division by zero for all x, logarithms of nonpositive values, etc.).
- Codomain compatibility: For a function f: R → R, the output f(x) must be a real number for every real x. So the expression must not produce non-real values for any real x.
- Simplicity and conventional form: The rule is often given as an algebraic expression, piecewise definition, etc., but any expression that yields real values for all real x is acceptable as a function from R to R, provided it is well-defined everywhere in the domain.

Analyzing the candidate f(x) = x^2 + 5:
- Domain check: For all real x, x^2 is defined and nonnegative, so x^2 + 5 is defined for every real x. There is no division by zero, no square roots of negative numbers, and no logarithms; hence it is well-defined on all of R.
- Codomain check: Since x^2 ≥ 0 for all x, x^2 + 5 ≥ 5, so the output is always a real number. Therefore f(x) maps reals to reals.
- Conclusion for the candidate: Based solely on these criteria, f(x) = x^2 + 5 is a valid rule for f: R → R.

If there were other options, we would similarly evaluate each one against these criteria: defineusability for all real x and yielding real outputs. Options that introduce undefined expressions for some x (like 1/(x-1) at x = 1) or outputs outside the real numbers would be invalid. Conversely, any expression that is a real-valued function for every real x would be valid.

In summary, given the information present, the expression x^2 + 5 is a valid choice for f(x) because it is defined for all real x and produces real outputs. If you can share the missing answer options, I can compare each one in turn and explain why each is valid or invalid relative to these criteria.

MATHS 108 Quiz 2

Consider the function f:R→R, with the rule f(x). Which of the following would be a valid choice for the rule f(x)?

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