Problem 52 - Algebra 2

Question

For which function is the range \(\\{y | y \leq 0\\} ?\) F. \(f(x)=-x\) G. \(f(x)=[x]\) H. \(f(x)=|x|\) J. \(f(x)=-|x|\)

Step-by-Step Solution

Answer

Function J, \(f(x) = -|x|\), has a range \(\{y | y \leq 0\}\).

Step 1: Understanding the Problem

We need to identify which given function has a range of all values less than or equal to 0, that is, \(\{y | y \leq 0\}\). This means any valid outputs must be negative or zero.

Step 2: Exploring Function F

Consider the function \(f(x)=-x\). For this function, for any value of \(x\), \(f(x)=-x\). The range includes positive values when \(x\) is negative, with no upper bound. Therefore, \(-x\) has a range that includes positive numbers, which doesn't fit our requirement of \(y \leq 0\).

Step 3: Exploring Function G

Consider the function \(f(x)=[x]\), where \([x]\) denotes the greatest integer less than or equal to \(x\). This function's range includes both negative and positive integers as \(x\) varies, which doesn't satisfy \(y \leq 0\) exclusively.

Step 4: Exploring Function H

Consider the function \(f(x)=|x|\). The absolute value function outputs values \(|x|\) that are always non-negative, meaning the range is \([0, \infty)\), which only provides non-negative values. This does not meet the condition \(y \leq 0\).

Step 5: Exploring Function J

Consider the function \(f(x)=-|x|\). The absolute value \(|x|\) is always positive or zero, and multiplying by -1 gives values \(-|x|\) that are always less than or equal to zero. Therefore, \(f(x)=-|x|\) outputs only non-positive values, fitting \(y \leq 0\).

Step 6: Conclusion

Among the options, the function \(f(x)=-|x|\) satisfies the condition that all outputs \(y \leq 0\). Thus, the correct choice is Function J.

Key Concepts

Absolute Value Function

When we talk about the absolute value function, denoted by \(|x|\), we refer to a mathematical function that provides the non-negative magnitude of a number or expression, without considering its sign. This function takes any real number as input and returns its distance from zero on the number line.

• If the number is positive, \(|x| = x\).
• If the number is negative, \(|x| = -x\).
• For zero, \(|x| = 0\).

Understanding the behavior of an absolute value function is essential, especially when identifying ranges and outputs. In particular, the output of \(|x|\) is always positive or zero. Therefore, if we want outputs that are \(y \leq 0\), the regular absolute value function won't suffice because its range is \([0, \infty)\), which includes only zero and positive values. But when the absolute value is negated, as in the function \(f(x) = -|x|\), this reverses the range, perfectly fitting outputs that meet the requirement of \(y \leq 0\).

Greatest Integer Function

The greatest integer function, represented as \([x]\), is another interesting mathematical function. This function is often called the "floor function." The main characteristic is that it rounds down to the nearest whole number less than or equal to the given number. Understanding its range is critical here.

• For positive inputs like 3.7, \([3.7]=3\).
• For negative numbers like -2.3, \([-2.3]=-3\), because -3 is the greatest integer less than -2.3.
• For integers like 5, it simply results in \([5]=5\).

The greatest integer function results in non-negative and negative whole numbers, so it covers all integers \(\mathbb{Z}\). This function doesn't provide outputs exclusively less than or equal to zero since it includes positive integers. Therefore, it doesn't fit the requirement \(y \leq 0\) for the range since it also includes outputs greater than zero.

Negative Values

Negative values are numbers less than zero and are an important part of understanding function ranges, especially under specific conditions such as those explored in this exercise. They are typically indicated by a minus sign.

• In terms of ordering, negative numbers fall to the left of zero on the number line.
• Common examples include \(-1\), \(-2\), and \(-0.5\).

A function whose outputs are restricted to be less than or equal to zero will invariably include negative values as part of its range. Concretely, the function \(f(x) = -|x|\) leverages the fact that the absolute value is always non-negative to ensure that its negation, \(-|x|\), is always zero or negative, perfectly matching \(y \leq 0\). Recognizing how a function limits its range through negative values can be an insightful exercise when analyzing or graphing such functions.