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Chapter 3: Problem 69

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 2|x|+y=0 $$

Short Answer

Expert verified
Yes, \( y = -2|x| \) defines \( y \) as a function of \( x \).

Step by step solution

01

Isolate the Variable y

First, we will solve the equation for \( y \). Start by subtracting \( 2|x| \) from both sides of the equation: \[ y = -2|x| \].
02

Identify the Type of Function

The equation \( y = -2|x| \) describes \( y \) as being directly affected by \( x \), specifically through the absolute value of \( x \). The absolute value function \( |x| \) affects how the function behaves by ensuring that any value of \( x \) is converted into a non-negative number.
03

Determine Function Criteria

For \( y \) to be a function of \( x \), each input \( x \) must produce exactly one output \( y \). The equation \( y = -2|x| \) guarantees that for every value of \( x \), there is exactly one corresponding value of \( y \). This confirms the definition of a function.

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Key Concepts

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

Absolute Value
Absolute value is an important mathematical concept that affects how functions behave. In the equation given, we see that the absolute value notation, denoted by vertical bars such as \(|x|\), ensures that the values are non-negative. This means whether \(x\) is positive or negative, \(|x|\) turns it into its positive counterpart.
  • If \(x = 3\), then \(|x| = 3\).
  • If \(x = -3\), then \(|x| = 3\).

This property is crucial in solving equations because it affects how we interpret the variable increases or decreases. Absolute value functions are often used to measure distances and are foundational in other mathematical operations.
Function Criteria
Understanding whether an equation defines a function is key in many mathematical studies. For an equation to define \(y\) as a function of \(x\), each input value of \(x\) must produce only one output value of \(y\). We refer to this as the function criteria.

The original equation, after solving for \(y\), becomes \(y = -2|x|\). Here, no matter what value you plug into \(x\), the result will produce a single value for \(y\). This satisfies the criterion for being a function. In simple terms, functions always produce predictable results鈥攁 requirement for consistent mathematical modeling.
Input-Output Relationship
The concept of input-output relationships in functions explains how inputs (changes in \(x\)) affect outputs (changes in \(y\)). In the equation \(y = -2|x|\), \(x\) is the variable input that guides the changes in \(y\).
  • If \(x = 1\), \(y = -2(1) = -2\).
  • If \(x = -1\), \(y = -2(-1) = -2\) (still \(-2\), due to the absolute value).

This relationship helps in understanding how altering the input influences the output. In this case, multiplying the absolute value by \(-2\) ensures that \(y\) decreases as \(|x|\) increases. This consistent pattern forms the core of understanding function behaviors. By recognizing these patterns, one can interpret complex mathematical relationships more easily.

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Most popular questions from this chapter

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 2 x+|y|=0 $$

\(13-16\) Draw the graphs of \(f, g,\) and \(f+g\) on a common screen to illustrate graphical addition. $$ f(x)=\sqrt[4]{1-x}, \quad g(x)=\sqrt{1-\frac{x^{2}}{9}} $$

Compositions of Odd and Even Functions Suppose that $$h=f \circ g$$ If \(g\) is an even function, is \(h\) necessarily even? If \(g\) is odd and \(f\) \(h\) odd? What if \(g\) is odd and \(f\) is odd? What if \(g\) is odd and \(f\) is even?

61鈥�68 ? Determine whether the function f is even, odd, or neither. If f is even or odd, use symmetry to sketch its graph. $$f(x) = 3x^{3} + {2x}^{2} + {1}$$

\(41-44\) Find \(f \circ g \circ h\) $$ f(x)=x-1, \quad g(x)=\sqrt{x}, \quad h(x)=x-1 $$
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