Question

Manipulating Functions.$$\text { If } x^{2}+y=x-2 y+3 x^{2}, \text { write } y=f(x)$$.

Short answer

The function of x represented as y is given by: \(y = f(x) = \frac{2}{3}x^2 + \frac{1}{3}x\).

Step-by-step solution

Move all terms involving y to one side

Subtract the term containing y from both sides to get the terms with y on one side of the equation: \[x^2 + y - (x - 2y) = 3x^2\]This simplifies to: \[x^2 + y - x + 2y = 3x^2\]Combine like terms: \[y + 2y = 3x^2 - x^2 + x\]Resulting in: \[3y = 2x^2 + x\]

Solve for y

Divide both sides of the equation by 3 to solve for y: \[\frac{3y}{3} = \frac{2x^2 + x}{3}\]This simplifies to: \[y = \frac{2x^2}{3} + \frac{x}{3}\]Now the equation represents y as a function of x, written as: \[y = f(x) = \frac{2}{3}x^2 + \frac{1}{3}x\]

Key concepts

Solving for y

When working with algebraic equations, solving for a particular variable means isolating that variable on one side of the equation. In the case of solving for y, our goal is to manipulate the equation so that y stands alone on one side, and everything else is on the other side. This process often involves a series of steps which can include distributing, combining like terms, and using inverse operations.

For example, in our exercise we start by rearranging the terms to get all y terms on one side. Once that's achieved, we combine the like y terms. Now, y is still being multiplied by 3, so the last step is dividing each side by 3, leaving y on its own. This process transforms the original equation into a function of x, denoted as f(x).

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic equations. Terms are considered like if they have the same variable raised to the same power. Adding or subtracting these like terms together makes the equation more manageable.

In our exercise, after some rearrangement, we have terms with y on the same side. We notice that y and 2y are like terms since they both have the variable y raised to the power of 1. By combining them, we reduce the equation to fewer terms, which simplifies the process of solving for y.

Functions in Mathematics

In mathematics, a function is a relationship between two sets of numbers or variables. For each element in the first set, called the domain, there is exactly one element in the second set, called the range. A function is often described by an equation, such as y = f(x), indicating that y is a function of x.

Functions can represent physical phenomena, economic trends, and more. They are a foundational concept in many areas of mathematics, from algebra to calculus. The process of manipulating an equation to write y as f(x) helps us to better understand the relationship between variables and to easily evaluate the function for different values of x.

Algebraic Equations

Algebraic equations are statements that represent the equality of two algebraic expressions. They consist of variables, constants, operations like addition, subtraction, multiplication, and division, and sometimes exponents.

To solve an equation means to find all the values of the variables that make the equation true. In our initial equation, we have a mix of x’s and y’s and we need to perform several steps to isolate y and express it in terms of x alone. Doing so involves understanding the properties of equality and knowing how to manipulate expressions methodically. The ability to solve algebraic equations is vital, as it applies to a wide range of disciplines within and beyond mathematics.