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Chapter 3: Problem 64

Without graphing, find the domain of each function. $$ f(x)=-3|x+5.7| $$

Short Answer

Expert verified
The domain of the function is all real numbers, \( (-\infty, \infty) \).

Step by step solution

01

Understand the Function

The function given is \( f(x) = -3|x + 5.7| \). This is a composition of a constant multiplication and an absolute value function. We need to determine the values of \( x \) for which this function is defined.
02

Determine Domain Constraints

An absolute value function like \( |x + 5.7| \) has no intrinsic restrictions like division by zero or taking the square root of a negative number. Thus, it is defined for all real numbers \( x \).
03

State the Domain

Since there are no restrictions arising from the absolute value operation and no other constraints in the function, the domain of \( f(x) = -3|x + 5.7| \) is all real numbers.

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Key Concepts

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

Domain of a Function
The domain of a function refers to all the possible input values (usually represented by \( x \)) that can be used in the function. For any function, determining the domain is crucial because it tells us the set of values for which the function is defined and yields valid outputs.
For example:
  • If a function involves division by a variable, we must exclude any values that would lead to division by zero.
  • If a function involves a square root, we only consider values that do not result in a negative number under the root sign.
In the case of the function \( f(x) = -3|x + 5.7| \), we look at the "absolute value" portion, \(|x + 5.7|\). Since absolute value functions are defined for all real numbers, there are no restrictions due to dividing by zero or square roots of negatives. Therefore, the function can accept any real number as input, and the domain is all real numbers.
Absolute Value
The concept of absolute value essentially measures the distance of a number from zero on the number line, without regard to direction. Absolute value is denoted by vertical bars: \( |x| \).
Key properties include:
  • \(|x|\) is always non-negative, meaning \(|x| \geq 0\) for any \(x\).
  • For any positive number \(a\), \(|a| = a\), and for any negative number \(b\), \(|b| = -b\).
  • \(|0| = 0\), since zero is at a zero distance from itself.
In the function \( f(x) = -3|x + 5.7| \), the absolute value \(|x + 5.7|\) helps maintain that the output of the expression inside the absolute value, \(x + 5.7\), is always treated as a non-negative value. Thus, regardless of whether \(x + 5.7\) is positive or negative, the absolute value converts it to a non-negative number, which is then scaled by \(-3\).
Real Numbers
Real numbers encompass a broad category of numbers that can represent a complete and continuous range on the number line. They include both rational numbers (like fractions and integers) and irrational numbers (numbers that cannot be expressed as a simple fraction).
Characteristics of real numbers:
  • They include integers like -2, 0, and 1.
  • They include decimals like 0.75 and -3.1415.
  • They cover numbers that can be represented as fractions such as \(\frac{1}{2}\).
  • They also include irrational numbers like \(\sqrt{2}\) and \(\pi\).
In the context of the function \( f(x) = -3|x + 5.7| \), the "real numbers" are important because the domain of the function is all real numbers. This means any real number can be substituted for \(x\), ensuring the function is well-defined across the entire real number spectrum.

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Most popular questions from this chapter

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