Question

Determine whether \(y\) is a function of \(x\). $$ x^{2} y-x^{2}+4 y=0 $$

Short answer

Yes, \(y\) is a function of \(x\), and it's defined as \(y=\frac{x^{2}}{x^{2}+4}\).

Step-by-step solution

Re-arrange the Equation to Isolate y

The equation right now is in form \(x^{2} y-x^{2}+4 y=0\). Start rearranging it by adding \(x^{2}\) to both sides of the equation which will give: \(x^{2} y+4 y=x^{2}\).

Factor y

To further isolate \(y\), factor it from the left-hand side of the equation. This yields: \(y(x^{2}+4) = x^{2}\).

Solve for y

Finally, divide both sides of the equation by \(x^{2}+4\) to solve for \(y\), giving the final equation: \(y=\frac{x^{2}}{x^{2}+4}\).

Key concepts

Function Concept

Understanding the 'function concept' is fundamental in mathematics, especially when analyzing relationships between variables. A function is a special type of relation where each input (x-value) produces exactly one output (y-value). This 'one-to-one' relationship ensures that for every x value, there is a single corresponding y value.

For example, when examining if y is a function of x, we look for instances where an x-value would yield more than one y-value. If this never happens, y is a function of x. In the given exercise, when y is expressed as \(y=\frac{x^{2}}{x^{2}+4}\)), we see that for every x-value input into the equation, a single y-value is produced, satisfying the definition of a function.

Solving Equations

Solving equations is a crucial skill in algebra and involves finding the value(s) of the unknown variable(s) that satisfy the equation. The steps taken in the provided solution serve as a classic example of problem-solving strategy: modifying the equation into a solvable form.

The process usually begins with simplifying terms, eliminating fractions, combining like terms, and moving terms to one side of the equation to isolate the variable sought. It is essential to apply the same mathematical operation to both sides of the equation to maintain equality. The equation from the exercise \(x^{2} y-x^{2}+4 y=0\)) is rearranged into a form where y can be solved, showing the solution's systematic approach.

Factoring

Factoring plays a key role in mathematics, particularly in simplifying expressions and solving equations. The term 'factoring' refers to breaking down a complicated expression into products of simpler expressions that, when multiplied together, give the original expression. It is often used as a step to solve for a variable.

In the exercise, \(y\)) is factored out from \(x^{2} y + 4 y\)), resulting in \(y(x^{2}+4)\)), demonstrating how factoring can help re-organize an equation to make the process of isolating a variable more straightforward.

Isolating Variables

The process of 'isolating variables' is a methodical approach to solving equations, wherein one variable is singled out on one side of the equation in order to find its value. Isolation is typically achieved by using basic algebraic operations such as addition, subtraction, multiplication, division, and factoring.

In the given problem, after factoring, we isolate y by dividing both sides of the equation by \(x^{2}+4\)). As a result, y alone stands on one side of the equation, clearly showing it as a function of x given by \(y=\frac{x^{2}}{x^{2}+4}\)). This illustrates the importance of isolating the variable in solving equations and reaffirms y's dependency on x.