Problem 14 - College Algebra Essentials

Question

Determine whether each equation defines \(y\) as a function of \(x .\) $$ x^{2}+y=25 $$

Step-by-Step Solution

Answer

Yes, the given equation \(x^{2}+y=25\) defines \(y\) as a function of \(x\).

Step 1: Rewrite the equation

Start by rewriting the original equation in the form \(y = f(x)\). Subtract \(x^{2}\) from both sides of the equation to get: \(y = 25 - x^{2}\)

Step 2: Analyze the equation

Next, analyze the equation \(y = 25 - x^{2}\). Here, for any real number \(x\), there will be one and only one value of \(y\). So \(y\) is a function of \(x\).

Key Concepts

Functions in Mathematics

In mathematics, the concept of a function is fundamental. Imagine it as a special type of relation between two sets, typically referred to as the domain and the range. For every element in the domain, a function assigns exactly one element in the range. Take an analogy of a vending machine: you select a product (input) and you get exactly one item (output).

Looking at our exercise, determining whether an equation defines a function is akin to checking if each input, in this case each value of \(x\), is linked to a singular output, which would be the corresponding value of \(y\). The equation \(y = 25 - x^2\) implies that for every input \(x\), there is a clear output \(y\), satisfying the very criterion of a mathematical function.

Algebraic Equations

Algebraic equations serve as the foundation for expressing mathematical relationships. They consist of variables and constants structured using mathematical operations such as addition, subtraction, multiplication, and division. The equation \(x^2 + y = 25\) is a classic representation of an algebraic equation involving two variables, \(x\) and \(y\).

The process of defining functions in algebra involves manipulating these equations to isolate one variable in terms of the others. In our step-by-step solution, we subtract \(x^2\) from both sides to express \(y\) solely as a function of \(x\), resulting in the equation \(y = 25 - x^2\). This process is crucial in algebra, as it transforms the equation into a form where the function's behavior becomes more discernible.

Function Analysis

Function analysis is a deeper dive into understanding the intricacies of functions. It involves looking at the characteristics of functions, such as their domain, range, continuity, and whether they are one-to-one or many-to-one. In analysing the given equation \(y = 25 - x^2\), one must note that every \(x\) value produces a unique \(y\) value. This unique correspondence signifies that the equation represents a function.

An important aspect of function analysis is graphing. For our particular function, if we were to graph it, we would see a parabola opening downwards, indicating that as \(x\) increases or decreases, \(y\) decreases after reaching a maximum value at \(y = 25\). Understanding this visual representation is critical for grasping the behavior of the function across its entire domain.