Question

True or False If no domain is specified for a function \(f,\) then the domain of \(f\) is the set of real numbers.

Short answer

False

Step-by-step solution

Define the default domain

If no domain is specified for a function, the default domain is typically considered to be the largest set of real numbers for which the function is defined. This means considering all real numbers except those that would make the function undefined.

Consider common exceptions

Common reasons a function may not be defined for all real numbers include division by zero and taking the square root of a negative number. These cases should be excluded from the domain.

Generalize the domain for common functions

For example, the function \( f(x) = \frac{1}{x} \) is not defined when \( x = 0 \), and the function \( f(x) = \sqrt{x} \) is not defined for negative values of \( x \). Therefore, the default domain would exclude such values.

Conclusion

Since not all functions can be defined for every real number, the statement 'If no domain is specified for a function \( f \), then the domain of \( f \) is the set of real numbers' is generally False.

Key concepts

undefined values

When studying functions, it's essential to understand that some values can make the function undefined. For example, consider a function like \( f(x) = \frac{1}{x} \). This fraction is undefined when \( x = 0 \), because dividing by zero is not possible. Another situation is the function \( f(x) = \sqrt{x} \), which is undefined for negative values of \( x \), as the square root of a negative number isn't a real number.

• Division by zero
• Square root of negative numbers

Thus, when identifying the domain, it's crucial to exclude these undefined values.

default domain

The default domain of a function refers to the largest possible set of input values for which the function is defined. When no specific domain is given, we generally consider all real numbers. However, we exclude values that would make the function undefined.
For instance, for the function \( f(x) = \frac{1}{x} \), the default domain would include all real numbers except zero, because the function is undefined at \( x = 0 \). Similarly, for \( f(x) = \sqrt{x} \), the default domain would include all non-negative real numbers, since the function is not defined for negative values.

• Default domain includes all real numbers
• Exclude values where the function is undefined

real numbers

In mathematics, real numbers include all the numbers that can be found on the number line. This includes both positive and negative numbers, zero, and fractions. However, some values within the set of real numbers might not work for certain functions.
For example, any number you can think of is a real number: 1, -5, 3.14, and even fractions like 1/2 and -3/4. But when we talk about the domain of a function, we sometimes need to exclude some real numbers, especially those that make the function undefined.

• Real numbers include positive, negative numbers, and zero
• All fractions are part of real numbers
• Not all real numbers can be part of a specific function's domain