Problem 25 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

Yes, the equation defines \(y\) as a function of \(x\).

Step 1: Rearrange the equation in terms of \(y\)

Start by rearranging the equation in order to express \(y\) in terms of \(x\). Rewrite the equation \(|x|-y=2\) into \(y=|x|-2\). Now bearing in mind that \(|x|\) is an absolute function and can break down into two parts, \(x\) for \(x ≥ 0\) and \(-x\) for \(x< 0\)

Step 2: Looking at the two scenarios

Investigate both scenarios: 1) \(x ≥ 0\) then the equation becomes \(y = x - 2\). 2) \(x< 0\) then the equation becomes \(y = -x - 2\). Both of these are linear equations, meaning for every \(x\) there is a unique \(y\) .

Step 3: Conclusion

Both equations correspond to the function definition where every value of \(x\) maps to only one unique value of \(y\). Thus, the initial equation indeed defines \(y\) as a function of \(x\).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in precalculus that assigns the non-negative magnitude of a real number, regardless of its sign. Expressed mathematically, the absolute value of a number 'x' is shown as \(|x|\) and is defined such that \(|x| = x \text{ if } x \text{ is greater than or equal to } 0 \text{ and } |x| = -x \text{ if } x \text{ is less than } 0\text{.}\)

When graphed on a coordinate plane, the absolute value function appears as a 'V' shape, where the point of the 'V' is known as the vertex. This distinctive shape reflects the function's ability to convert all inputs to non-negative outputs. For example, \(|-3| = 3\text{ and } |3| = 3\text{.}\)

Understanding this function is crucial in solving the given exercise, because handling the equation \(|x| - y = 2\text{ requires considering two separate scenarios based on the sign of } x\text{.}\)

Linear Equations

Linear equations are algebraic expressions that represent straight lines when plotted on a graph. The standard form of a linear equation is \(y = mx + b\text{,}\) where 'm' is the slope and 'b' is the y-intercept. The slope determines the angle at which the line tilts, and the intercept is the point where the line crosses the y-axis.

In our exercise, when breaking down the absolute value function into two parts, we derive two linear equations: for \(x \text{ greater or equal to } 0\text{, it is } y = x - 2\text{,}\) while for \(x \text{ less than } 0\text{, it is } y = -x - 2\text{.}\) These equations describe straight lines and each value of 'x' corresponds to exactly one value of 'y', a fundamental characteristic of a function. It's important to notice how finding the value of 'y' for a given 'x' is straightforward, aiding our understanding of how each part of the original equation behaves over its respective domain of ‘x’ values.

Function Definition

In mathematics, a function is a relation that uniquely associates members of one set (the domain) with members of another set (the range). Simply put, every 'input' or 'x-value' corresponds to exactly one 'output' or 'y-value'. This one-to-one relationship is fundamental to the concept of a function.

The given exercise revolves around determining if the equation \(|x| - y = 2\text{ represents 'y' as a function of } x\text{.}\) By rearranging the equation to become \(y = |x| - 2\text{,}\) and then considering the two conditions inherent to the absolute value function, we obtain two linear equations that fulfill the function definition: every 'x' has just one 'y'. The function is defined for all real numbers ‘x’, showcasing how even a piecewise scenario like the absolute value function maintains the integral quality of a function, ensuring that for every 'x' in the domain, we can find a single 'y' in the range.