Problem 11 - College Algebra

Question

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). $$ 3 x^{2}+y=14 $$

Step-by-Step Solution

Answer

Yes, it represents \(y\) as a function of \(x\).

Step 1: Understanding the Equation

The given equation is \(3x^2 + y = 14\). Our task is to determine if \(y\) can be written explicitly as a function of \(x\). This means we need to check if for each \(x\), there is exactly one corresponding \(y\) value.

Step 2: Solve for y

To see if \(y\) is a function of \(x\), we try to express \(y\) in terms of \(x\). First, isolate \(y\) by subtracting \(3x^2\) from both sides to get \(y = 14 - 3x^2\). This expresses \(y\) as a function of \(x\), often written as \(y = f(x)\).

Step 3: Verify Function Criteria

In the rewritten equation \(y = 14 - 3x^2\), for any given \(x\), there is exactly one value for \(y\). This satisfies the definition of a function, which requires that each input \(x\) maps to exactly one output \(y\).

Key Concepts

Determining Functions

To figure out if a relation is a function, we need to see if each input variable, commonly represented as \(x\), corresponds to exactly one output \(y\). This is what makes it a function. In simpler terms, for every value you plug in as \(x\), you should only get one unique result for \(y\).

Consider an equation from your math textbook, like \(3x^2 + y = 14\). Our goal here is to see if this equation can be rearranged so that \(y\) is by itself on one side. When \(y\) is isolated, it's easier to see if \(y\) is explicitly dependent on \(x\). This exemplifies determining a function, where we evaluate whether changing \(x\) uniquely determines \(y\).

The outcome of this process allows us to either confirm or deny that a relation illustrates a function of \(x\). By isolating \(y\), we make the function explicit and understandable.

Equation Manipulation

Equation manipulation is a crucial skill in mathematics. It helps us rearrange equations to better understand them or solve for one variable in terms of others. For our equation \(3x^2 + y = 14\), manipulating the equation involves subtracting \(3x^2\) from both sides to isolate \(y\).

Here's how it's done step by step:

• Original equation: \(3x^2 + y = 14\)
• Subtract \(3x^2\) from both sides: \(y = 14 - 3x^2\)

Once this manipulation is complete, \(y\) is clearly expressed as a function of \(x\), which facilitates further analysis.

This step is like rearranging your room to better understand what space is available and how everything fits together. It makes it clear how \(y\) responds to changes in \(x\) and simplifies verifying whether our original relation is indeed a function.

Criteria for Functions

A relationship between \(x\) and \(y\) is classified as a function when each \(x\) associates with a single \(y\). This is the fundamental criterion.

In our equation \(y = 14 - 3x^2\), for each different \(x\), there is exactly one corresponding \(y\) value. This one-to-one correspondence is a definitive characteristic of a function. The process of verifying this ensures we can confidently describe the relationship as a function.

Here's a quick recap of the criteria that must be met:

• Each input \(x\) maps to only one output \(y\).
• The relation must make it impossible for any single \(x\) to correspond to multiple \(y\)'s.

Our successful manipulation of the equation to \(y = 14 - 3x^2\) demonstrates that this relation fulfills all necessary criteria for a function. This understanding is essential in mathematics as it lays the groundwork for more complex function-related topics.