f(t+1) = 3(t^2 + 2t + 1) + 5(t+1) + 2 - Treasure Valley Movers

Understanding and Simplifying the Function: f(t+1) = 3(t² + 2t + 1) + 5(t+1) + 2

Mathematics often presents functions in complex forms that may seem intimidating at first glance, but breaking them down clearly reveals their structure and utility. In this article, we’ll explore the function equation:

f(t+1) = 3(t² + 2t + 1) + 5(t+1) + 2

We’ll simplify it step-by-step, interpret its components, and explain how this function behaves—and why simplifying helps in solving equations, graphing, or applying it in real-world contexts.

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Step 1: Simplify the Right-Hand Side

We begin by expanding and combining like terms on the right-hand side of the equation:

f(t+1) = 3(t² + 2t + 1) + 5(t + 1) + 2

Step 1.1 Expand each term:

• 3(t² + 2t + 1) = 3t² + 6t + 3
  
• 5(t + 1) = 5t + 5

Step 1.2 Add all expanded expressions:

f(t+1) = (3t² + 6t + 3) + (5t + 5) + 2

Step 1.3 Combine like terms:

• Quadratic: 3t²
  
• Linear: 6t + 5t = 11t
  
• Constants: 3 + 5 + 2 = 10

So,
f(t+1) = 3t² + 11t + 10

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Step 2: Understand What f(t+1) Means

We now have:

f(t+1) = 3t² + 11t + 10

This form shows f in terms of (t + 1). To find f(u), substitute u = t + 1, which implies t = u − 1.

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Step 3: Express f(u) in Terms of u