Problem 68 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 10.) $$ \sqrt{x}+y=12 $$

Step-by-Step Solution

Answer

Yes, \(y = 12 - \sqrt{x}\) defines \(y\) as a function of \(x\).

Step 1: Isolate y

Start by isolating the term that contains \(y\) to express \(y\) explicitly in terms of \(x\). This involves subtracting \(\sqrt{x}\) from both sides of the equation. The resulting equation is: \[ y = 12 - \sqrt{x} \]

Step 2: Verify Function Definition

To determine whether \(y\) is a function of \(x\), recall that a function is a relation in which each input value must correspond to exactly one output value. Check the expression \(y = 12 - \sqrt{x}\): For every value of \(x\), there is exactly one resulting value of \(y\) because subtraction of a square root value results in only one value. Thus, \(y\) does define a function of \(x\).

Key Concepts

Function Definition

A function in algebra is a special type of relation. It describes a process where you input a value and get exactly one output. Think of it like a vending machine. You put in a selection (input), and you always receive the same snack (output). This is what makes the concept of a function critical in algebra. There's a key rule to remember. Each 'x' (input) in a function can lead to only one 'y' (output). This uniqueness is what separates functions from other mathematical relations. It ensures consistency and predictability in our calculations. Whether or not an equation defines a function can often be visualized by the vertical line test when graphed. However, in algebra, we often determine functions by examining the equation itself. Always ask yourself if there is one and only one output for each input.

Isolating Variables

When working with algebraic equations, isolating variables is a common technique. This means arranging the equation in such a way that one variable stands on one side of the equation by itself. For example, when given the equation \(\sqrt{x}+y=12\), we isolate \(y\) to determine if it's a function of \(x\).**Steps to Isolate Y:**

• Identify the variable you need to isolate. Here it's \(y\).
• Use basic algebraic operations: addition, subtraction, multiplication, or division.
• Subtract \(\sqrt{x}\) from both sides to get \(y = 12 - \sqrt{x}\).

The end result is an equation where \(y\) is neatly singled out. This makes it explicit that \(y\) is expressed purely in terms of \(x\). Isolating the variable often makes it easier to test whether an equation defines a function.

Function Verification

Verifying that an equation defines a function involves checking to make sure each input has a unique output. After isolating variables, we need to ensure that every value of \(x\) produces one and only one output value for \(y\).**Why is it a function?**- The expression \(y = 12 - \sqrt{x}\) ensures that for every value of \(x\) within the domain, \(y\) will have a single corresponding outcome.- Think about the subtraction process: no matter what positive value of \(x\) you plug in, you end up subtracting a specific square root, leading to one distinct value for \(y\).In algebra, being confident that every input maps to one unique output is crucial for recognizing functions. This guarantee allows us to predict outcomes reliably and simplifies mathematical modeling.