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Chapter 2: Problem 39

Determine whether each equation defines \(y\) as a function of \(x\). $$y+2=|x|$$

Short Answer

Expert verified
Yes, y is a function of x.

Step by step solution

01

Rewrite the Equation

Start by isolating the variable y in the equation. Given the equation: \[y + 2 = |x|\]Subtract 2 from both sides of the equation to get: \[y = |x| - 2\]
02

Check for Function Definition

An equation defines y as a function of x if every value of x corresponds to exactly one value of y. Let's analyze the transformed equation\[y = |x| - 2\]The absolute value function \(|x|\) always produces a non-negative output for any real number x, making \(|x| - 2\) a unique output for each input x.
03

Conclusion

Since the equation \(y = |x| - 2\) produces a unique value of y for every real number x, it defines y as a function of x.

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Key Concepts

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

Absolute Value Functions
Absolute value functions are an important concept in precalculus. The absolute value of a number, denoted by \(|x|\), is its distance from zero on a number line, regardless of direction. This means it's always non-negative. For example, \(|-3| = 3\) and \(|3| = 3|\), since both are three units away from zero.

In the given equation \(|x| - 2|\), the absolute value ensures that \(|x|\) is always zero or positive. When we subtract 2, the result can be any real number. This property is crucial in determining function behavior, as seen in the exercise.
Function Definition
A function is a mathematical relationship where each input (usually x) corresponds to exactly one output (usually y). If each x value maps to a single y value, it's a function. This concept ensures consistent results, avoiding multiple outputs for a single input.

In the exercise, the equation \(|y = |x| - 2|\) correctly defines y as a function of x. Every x yields a unique y, fulfilling the function criteria.

To check this, observe that \(|x|\) handles any input x, returning a specific non-negative output that, when adjusted by -2, results in a unique y value. Hence, the equation identifies a consistent relationship between x and y.
Real Numbers
Real numbers encompass all the numbers on the number line, including rational (fractions, finite/infinite decimals) and irrational numbers (like \(\pi\) or \(\sqrt{2}\)). Real numbers can be positive, negative, or zero.

In the context of the exercise, when x is any real number, the absolute value function \(|x|\) processes x and produces a non-negative result. By subtracting 2 from this result, we get a corresponding output y.

This interaction within the realm of real numbers ensures the equation \(|y = |x| - 2|\) defines y as a function of x across the entirety of the real number set.

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Most popular questions from this chapter

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which \(f\) is increasing, decreasing, or constant. $$f(x)=\left\\{\begin{array}{ll} \sqrt{-x} & \text { for } x<0 \\ -\sqrt{x} & \text { for } x \geq 0 \end{array}\right.$$

Solve each problem. The cost in dollars of shipping a machine is given by the function \(C=200+37[w / 100]\) for \(w>0\) where \(w\) is the weight of the machine in pounds. For which values of \(w\) is the cost less than \(\$ 862 ?\)

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=x^{6}-x^{4}+x^{2}$$

Use transformations to graph each function and state the domain and range. $$y=-\sqrt{x-3}+1$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=|x|-9$$
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