Problem 13 - Precalculus Essentials

Question

Determine whether each equation defines y as a function of \(x .\) $$x^{2}+y=16$$g

Step-by-Step Solution

Answer

Yes, the equation \(x^{2}+y=16\) defines y as a function of x as any x-value will have a unique corresponding y-value.

Step 1: Rearrange the Equation

Firstly, rearrange the equation \(x^{2}+y=16\) to isolate y. This is done by subtracting \(x^{2}\) from both sides of the equation. This results in the equation: \(y=16-x^{2}\) .

Step 2: Examine the Equation

The equation \(y=16-x^{2} \) shows that for any x value input, there will be only one corresponding y value. This is because any number squared will produce only one result, i.e., every real number has only one square. Subtracting that result from 16 will also only produce one result, hence for any input x, there is only a single output y.

Key Concepts

Understanding a Function of x

In precalculus, the concept of a 'function of x' is fundamental. It represents a specific kind of rule or association between two sets of numbers, commonly referred to as the input and output. A function assigns to each input value exactly one output value, creating a unique pairing. The notation used is typically 'f(x)', where 'f' is the function and 'x' is the input variable. Functions can model relationships in real-world scenarios, answer questions or make predictions based on input data.

For instance, when we say 'y' is a function of 'x', we imply that 'y' changes in response to 'x'. In the given equation, \(y = 16 - x^{2}\), 'y' is indeed a function of 'x' because for every value of 'x' that we plug into the equation, there will be exactly one corresponding value of 'y'. No matter what value 'x' takes, squaring it and subtracting the result from 16 will yield one specific number for 'y'. This predictable relationship makes it a function. Functions are versatile tools in maths that allow us to encapsulate a variety of dynamic processes into a single equation.

Isolating Variables

The process of isolating a variable is a crucial algebraic technique. It involves manipulating an equation in such a way that the variable of interest is left on one side of the equation, and all other terms are moved to the opposite side. This technique is key to solving equations and understanding how changes in one variable affect another.

For example, in the problem at hand, we started with the equation \(x^{2} + y = 16\) and isolated 'y' by subtracting \(x^{2}\) from both sides. The resulting equation, \(y = 16 - x^{2}\), makes it much easier to see how 'y' depends directly on 'x'. Isolating variables can also help in graphing functions, predicting outcomes, and simplifying complex formulas. Remember, the goal is to 'untangle' the equation such that the variable you're solving for stands alone and becomes the subject of the formula.

Squaring Numbers

In mathematics, squaring a number refers to multiplying the number by itself. The 'square' of a number 'x' is written as \(x^{2}\). It is an important operation and comes up frequently in algebra, geometry, and many areas of advanced mathematics. When you square any real number, the result is always non-negative, because the product of two positive numbers or two negative numbers is positive.

Understanding squaring is essential for solving equations, analyzing curves, and working in many scientific fields. In the context of our equation, \(x^{2}\) signifies that each 'x' value will be squared, ensuring that the result contributing to the function's output will always be positive or zero. Moreover, the act of squaring numbers plays a pivotal role in the concept of functions because it can affect the shape and characteristics of the graph produced by a function, such as opening upward or downward and having a vertex, when it is part of a quadratic function like the one in our exercise.