91Ó°ÊÓ

In algebra, a function is a rule that assigns each input exactly one output. We often write functions as equations, like \( f(x) = 2x + 3 \). Here, if we input a value for \(x\), we can calculate the corresponding \(f(x)\). Functions help us understand relationships between variables.

A common example is linear functions, but functions can also be quadratic, polynomial, or even more complex.

In our exercise, we start with two functions:
1. \(y = 3t^2 - 3\)
2. \(t = x - 1\)

We need to write \(y\) as a function of \(x\). This means expressing \(y\) using only \(x\). This often involves substitution and simplification steps, which we will explore further.

The substitution method is a powerful tool for solving equations and expressing variables in terms of others. Here’s how it works:

1. Identify the equations or functions. We have \(y = 3t^2 - 3\) and \(t = x - 1\).
2. Express one variable in terms of the other variable. From \(t = x - 1\), \(t\) is already in terms of \(x\).
3. Substitute this expression into the other function. We replace \(t\) in \(y = 3t^2 - 3\) with \(x - 1\).

This gives us:
\(y = 3(x - 1)^2 - 3\)
Now, \(y\) is expressed entirely in terms of \(x\).

Substitution helps us transform our functions step-by-step, making complex relationships simpler. It’s especially useful when dealing with multiple equations or functions.

Simplifying expressions is about making an equation as straightforward as possible. After substituting, we need to simplify \(y = 3(x - 1)^2 - 3\).

Here’s how we do it:
1. Expand \((x - 1)^2\) to \(x^2 - 2x + 1\).
2. Substitute back into \(y\), giving us \(y = 3(x^2 - 2x + 1) - 3\).
3. Distribute the 3: \(y = 3x^2 - 6x + 3\).
4. Subtract 3: \(y = 3x^2 - 6x + 3 - 3\).
5. Simplify the constants: \(y = 3x^2 - 6x\).

Simplifying makes the function cleaner and easier to understand. Each step removes complexity, providing a final simplified function: \(y = 3x^2 - 6x\). This process is crucial in algebra as it turns complicated expressions into clear, manageable forms.