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Chapter 2: Problem 24

In Exercises 1鈥�30, find the domain of each function. $$ f(x)=\sqrt{84} $$

Short Answer

Expert verified
The domain of the function \( f(x) = \sqrt{84} \) is all real numbers, \( (-\infty, +\infty) \)

Step by step solution

01

Identify the Type of Function

The function \( f(x) = \sqrt{84} \) is not dependent on the variable \( x \) as the \( x \) is not present in the expression. Therefore, it is a constant function. Constant functions can accept any real number as an input.
02

Define the Domain of a Constant Function

The domain of a constant function is all real numbers, which is represented as \( (-\infty, +\infty) \)
03

Finalize the Domain

Hence, the domain of the function \( f(x) = \sqrt{84} \) is all real numbers, \( (-\infty, +\infty) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

constant function
A constant function is quite simple yet fundamental in mathematics. It is a type of function where, no matter what input you give, the output remains the same. Imagine that you have a vending machine. You can press any button, but it will always give you a soda, regardless of your choice. This function behaves similarly.
This is because the expression defining the function does not contain any variable. For example, in the function \( f(x) = \sqrt{84} \), the absence of \( x \) means that the operation \( \sqrt{84} \) is performed once, yielding a constant result.
Some key points about constant functions:
  • They have horizontal lines as graphs on the Cartesian plane, meaning the y-coordinate remains the same for any x-coordinate.
  • Their output is always fixed, irrespective of the input value.
  • These functions are easy to differentiate, as their derivative is zero due to no change occurring.
real numbers
Real numbers are a cornerstone of mathematics and fundamentally comprise both rational and irrational numbers. These include all the numbers on the number line, such as whole numbers, fractions, and decimals. Real numbers are used in measuring quantities and can be necessary for various aspects of daily life, from counting objects to dividing a pizza.
Some features of real numbers include:
  • Continuity: There is no gap between numbers on the real number line.
  • Infinite: Real numbers extend infinitely in both positive and negative directions.
  • Includes integers, fractions, and irrational numbers: Integers, fractions, and numbers like \( \pi \) and \( \sqrt{2} \) are all real numbers.
Understanding real numbers is crucial when determining the domain of functions because they form the broadest category of numbers that functions can accept as input, barring any specific restrictions imposed by the function.
domain of a function
The domain of a function is essentially the collection of all possible input values (usually x-values) that a function can accept. For example, when you find the domain of a function, you're essentially answering the question: "What values can I plug into this function without causing any mathematical mishaps?"
Different types of functions have different domains:
  • For polynomial functions: The domain is all real numbers, \((-\infty, +\infty)\) because there's no restriction for their inputs.
  • For fractions: The denominator cannot be zero, so we exclude those x-values that make the denominator zero from the domain.
  • For square roots: The expression inside must be non-negative to ensure real number results.
  • For constant functions like \( f(x) = \sqrt{84} \): The domain is always \((-\infty, +\infty)\), as they can take any real number as input, with no dependence on \( x \).
Knowing the domain of a function is crucial as it helps avoid undefined expressions, particularly when working with fractions or roots. It ensures that you are working within the "safe" values that maintain function definition.

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Most popular questions from this chapter

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation鈥檚 domain and range. $$(x+2)^{2}+(y+2)^{2}=4$$

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What is a circle? Without using variables, describe how the definition of a circle can be used to obtain a form of its equation.

Solve each quadratic equation by the method of your choice. $$-x^{2}-2 x+1=0$$

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