Problem 21 - Precalculus with Limits

Question

In Exercises 19-36, determine whether the equation represents \(y\) as a function of \(x\). \(x^2 + y = 4\)

Step-by-Step Solution

Answer

Yes, the equation \(x^2 + y = 4\) represents \(y\) as a function of \(x\).

Step 1: Rewrite the given equation

Start by rewriting the equation isolating \(y\) in terms of \(x\). This can be done by subtracting \(x^2\) from both sides of the equation. So, the equation will become \(y = 4 - x^2\).

Step 2: Analyze the Function

Examine the equation \(y = 4 - x^2\). For every real number \(x\), there exists exactly one value for \(y\). This means that the equation meets the condition to be a function, where for every \(x\) input there is a single \(y\) output.

Key Concepts

Function of x

Understanding what a function of x means is crucial in mathematics, as it represents a relationship where each input (x) is associated with exactly one output. In the equation from the exercise, y = 4 - x^2, we see that for every value of x that we can choose, there is a single, well-defined result for y. This characteristic is what defines a function.

For instance, if we input 2 for x, the output y will only be 4 minus 2 squared, which is 0. No matter how many times we perform this calculation, the outcome will remain the same as long as the input is 2. This predictability and uniqueness of the output signify that y is indeed a function of x. It's important to recognize that functions can take many forms, such as linear, quadratic, polynomial, exponential, and so on. The given equation represents a specific type called a quadratic function, reflected by the x^2 term.

Isolating Variables

Isolating variables is a method used to manipulate equations so that one variable stands alone on one side of the equation. This technique is essential for solving equations and understanding how the variables relate to each other. In our given exercise, we isolate y by moving x^2 to the other side of the equation through subtraction.

Here's how it's done: starting with x^2 + y = 4, we subtract x^2 from both sides to get y = 4 - x^2. Isolating variables allows us to examine the relationship between x and y more clearly as it highlights the dependence of y on x. It's a fundamental skill in algebra that enables students to solve for one variable in terms of the others, paving the way to graphing functions, solving systems of equations, and more.

Function Analysis

Function analysis involves examining the properties and behaviors of functions. It includes determining whether a certain relation qualifies as a function by checking if every input has a unique output, which is what we did in the exercise. The relation y = 4 - x^2 shows that for every value of x, there is only one possible value for y.

During the analysis, we also look at attributes such as the function's domain (all the possible x-values), range (all the possible y-values), intercepts, increasing or decreasing intervals, and any symmetries or asymptotes. By understanding these characteristics, one can predict how the function behaves without graphing it. Additionally, function analysis helps in finding the roots or zeros of the function (where the function crosses the x-axis) and in identifying maximum or minimum points, which are valuable in various applications such as physics, economics, and engineering.