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Chapter 2: Problem 54

Determine the domain and range of each relation. $$x=-|y|$$

Short Answer

Expert verified
Domain: All real numbers. Range: x 鈮� 0.

Step by step solution

01

Identify the Relation Format

The given relation is expressed as an equation: \[ x = -|y| \]. Here, the variable x is defined in terms of y using the absolute value function.
02

Determine the Domain

For the domain, identify all possible values of y for which the equation is defined. Since the absolute value function \(|y|\) is always non-negative, and there are no restrictions on y, y can be any real number. Therefore, \[ y \text{ can be } (-\text{infinity}, +\text{infinity}) \text{ or } y \text{ is in } \text{ the set of all real numbers }\textbf{R}. \]
03

Solve for the Range

To find the range, determine all possible values of x that result from substituting the domain values in the relation. Note that \(|y|\) is always positive or zero, so \[ -|y| \text{ will produce values } \text{ less than or equal to zero } \text{(i.e., } \text{ negative or zero values). \text{Thus, } \] \[ x \text{ can be } ({-\text{infinity}, 0] }. \]

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Key Concepts

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

Understanding Absolute Value Function
The absolute value function is a key mathematical concept often denoted as \(|y|\). This function measures the distance a number is from zero on the number line, without considering direction. Essentially, any number put into an absolute value function becomes non-negative.
For example:
  • \(|3| = 3\)
  • \(|-3| = 3\)
This property is useful in various mathematical contexts, especially when analyzing equations like \(x = -|y|\). Here, \(|y|\) ensures we consider only non-negative values, which are then negated in the equation. This creates a range of possible values for \(x\) that are all non-positive numbers.
Exploring Real Numbers
Real numbers include all the numbers you can think of on the number line. They encompass:
  • Natural numbers (1, 2, 3, ...)
  • Whole numbers (0, 1, 2, ...)
  • Integers (..., -2, -1, 0, 1, 2, ...)
  • Rational Numbers (fractions and decimals such as \(\frac{1}{2}\), 0.75)
  • Irrational Numbers (decimals that cannot be expressed as fractions such as \(\frac{\fname}{n}\))
When we talk about the domain in the context of the equation \(x = -|y|\), we are identifying all real numbers that \(y\) can be. Since there are no restrictions on \(y\) in this equation, \(y\) can be any real number, which means its domain is all real numbers. This simplifies understanding and interacting with complex equations.
Inequality Solutions in Math
Inequalities are mathematical statements that use symbols to compare the relationship between values.
For instance, in the solution to \(x=-|y|\), the inequality plays a pivotal role in describing the range of possible values for \(x\).
Here鈥檚 how:
  • The absolute value function \(|y|\) produces non-negative outcomes (zero or positive numbers).
  • Multiplying by -1 reverses the value to non-positive numbers (zero or negative numbers).
Hence, the resulting range of \(x\) values is represented as \([-\infty, 0]\). This signifies that \(x\) can take on any value from negative infinity up to and including zero. Understanding such inequalities aids in deciphering the constraints and possibilities in math problems.

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Most popular questions from this chapter

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=(x+3)^{2}$$

Solve each problem. The monthly water bill in Hammond is a function of the number of gallons used. The cost is \(\$ 10.30\) for \(10,000\) gal or less. Over \(10,000\) gal, the cost is \(\$ 10.30\) plus \(\$ 1\) for each 1000 gal over \(10,000\) with any fraction of 1000 gal charged at a fraction of \(\$ 1 .\) On what interval is the cost constant? On what interval is the cost increasing?

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=|x-2|$$

Use transformations to graph each function and state the domain and range. $$y=-2 \sqrt{x+3}+2$$

Solve each inequality by graphing an appropriate function. State the solution set using interval notation. $$5-\sqrt{x} \geq 0$$
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