Finding G(2) A Step-by-Step Guide To Piecewise Functions

Jul 24, 2025 by ADMIN 57 views

Decoding g(2): A Step-by-Step Solution to Finding the Value of a Piecewise Function

Hey guys! Today, we're diving into a fun little math problem involving a piecewise function. Don't let the name intimidate you; it's simpler than it sounds. We're going to break down the problem step by step, making sure everyone understands the logic behind finding the value of g(2). So, grab your thinking caps, and let's get started!

The Challenge: Finding g(2) in a Piecewise Function

Our main goal here is to determine the value of $g(2)$ given the function:

$g(x)=\left\{\begin{array}{ll} \left(\frac{1}{2}\right)^{x-3}, & x<2 \\ x^3-9 x^2+27 x-25, & x \geq 2 \end{array}\right.$

This might look a bit complex at first glance, but it's essentially a function that behaves differently depending on the value of $x$ . Let's dissect what this means and how we can find $g(2)$ .

Understanding Piecewise Functions

So, what exactly is a piecewise function? Think of it as a function that's been split into pieces, each with its own set of rules. Each piece is defined for a specific interval of $x$ values. In our case, $g(x)$ has two pieces:

When $x$ is less than 2 ( $x < 2$ ), the function is defined as $\left(\frac{1}{2}\right)^{x-3}$ .
When $x$ is greater than or equal to 2 ( $x \geq 2$ ), the function is defined as $x^3-9 x^2+27 x-25$ .

The key to solving these problems is knowing which piece of the function applies to the specific $x$ value we're interested in. In our case, we want to find $g(2)$ , so we need to figure out which rule to use when $x = 2$ .

Choosing the Right Piece

Now comes the crucial step: selecting the correct piece of the function. Since we're looking for $g(2)$ , we need to see which condition $x = 2$ satisfies. Looking at our function definition, we see that the second piece applies when $x \geq 2$ . This means that when $x$ is exactly 2, we use the rule:

$g(x) = x^3-9 x^2+27 x-25$

This is super important! We can't use the first piece because it only applies when $x$ is strictly less than 2. Using the wrong piece would lead to the wrong answer, and we definitely want to avoid that.

Plugging in the Value

Alright, we've identified the correct piece of the function. Now, it's just a matter of substituting $x = 2$ into the expression:

$g(2) = (2)^3 - 9(2)^2 + 27(2) - 25$

This looks like a bit of arithmetic, but don't worry, we'll take it one step at a time. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Crunching the Numbers

Let's break down the calculation:

Calculate the exponent: $(2)^3 = 2 * 2 * 2 = 8$ and $(2)^2 = 2 * 2 = 4$
Perform the multiplications: $9(4) = 36$ and $27(2) = 54$

Now our equation looks like this:

$g(2) = 8 - 36 + 54 - 25$

Perform the addition and subtraction from left to right:
- $8 - 36 = -28$
- $-28 + 54 = 26$
- $26 - 25 = 1$

So, after all that calculation, we arrive at our answer:

$g(2) = 1$

The Answer and Its Significance

Therefore, the value of $g(2)$ for the given piecewise function is 1. This means that when we input $x = 2$ into the function, the output is 1. Understanding how to evaluate piecewise functions is a fundamental concept in mathematics, and it pops up in various areas like calculus, discrete mathematics, and even computer science.

Why This Matters: The Importance of Piecewise Functions

You might be wondering, “Okay, we found g(2), but why do we even care about piecewise functions?” That's a valid question! Piecewise functions aren't just abstract mathematical concepts; they're actually quite useful in modeling real-world situations. They allow us to represent scenarios where the relationship between variables changes depending on the specific conditions.

Real-World Applications

Think about situations like:

Tax brackets: The amount of tax you pay often depends on your income level. Different income ranges are taxed at different rates, making it a perfect example of a piecewise function.
Shipping costs: The cost of shipping a package might depend on its weight. There might be a flat fee for packages under a certain weight, and then a per-pound charge for heavier packages.
Utility bills: The cost of electricity or water might vary depending on your usage. You might have a lower rate for the first certain amount of usage and a higher rate for usage beyond that.

In each of these cases, a single formula couldn't accurately represent the relationship between the variables. That's where piecewise functions come in handy. They provide the flexibility to model these situations with accuracy.

Beyond the Basics

Furthermore, piecewise functions are crucial in more advanced mathematical concepts. For example, they play a significant role in:

Calculus: Piecewise functions can be used to define functions that have different derivatives or integrals in different intervals. This is important in modeling situations where rates of change vary over time or space.
Computer science: Piecewise functions are used in programming to create conditional logic. For example, a program might perform different actions based on the value of a variable, which can be represented using a piecewise function.

Practice Makes Perfect: Tackling More Piecewise Function Problems

Now that we've successfully navigated our first piecewise function problem, it's time to solidify our understanding with a bit more practice. The more you work with these functions, the more comfortable you'll become with identifying the correct piece and plugging in values.

Example Problem 1

Let's consider another piecewise function:

$f(x) = \begin{cases} x^2 + 1, & x < 0 \\ 3x - 2, & 0 \leq x < 2 \\ 5, & x \geq 2 \end{cases}$

What is the value of $f(1)$ ?

Step 1: Identify the correct piece. Since $1$ falls within the range $0 \leq x < 2$ , we use the second piece: $f(x) = 3x - 2$ .
Step 2: Substitute the value. Plug in $x = 1$ : $f(1) = 3(1) - 2$ .
Step 3: Simplify. $f(1) = 3 - 2 = 1$ .

So, $f(1) = 1$ .

Example Problem 2

Now, let's try a slightly different one:

$h(x) = \begin{cases} -2x + 4, & x \leq -1 \\ x^3, & -1 < x < 1 \\ 2x - 1, & x \geq 1 \end{cases}$

What is the value of $h(-1)$ ?

Step 1: Identify the correct piece. Here, $x = -1$ satisfies the condition $x \leq -1$ , so we use the first piece: $h(x) = -2x + 4$ .
Step 2: Substitute the value. Plug in $x = -1$ : $h(-1) = -2(-1) + 4$ .
Step 3: Simplify. $h(-1) = 2 + 4 = 6$ .

Therefore, $h(-1) = 6$ .

Tips for Success

As you tackle more problems, keep these tips in mind:

Pay close attention to the conditions. The inequalities defining each piece are crucial. Make sure you choose the correct piece based on the given $x$ value.
Substitute carefully. Once you've identified the right piece, double-check that you're plugging in the value correctly.
Simplify step-by-step. Avoid making careless arithmetic errors by breaking down the calculation into smaller steps.

Conclusion: Mastering Piecewise Functions

Piecewise functions might seem a bit daunting at first, but as we've seen, they're quite manageable once you understand the underlying concept. By carefully selecting the correct piece based on the given $x$ value and performing the necessary calculations, you can confidently find the value of the function. Remember, practice is key! The more you work with piecewise functions, the more comfortable and proficient you'll become. Keep practicing, keep exploring, and you'll be a piecewise function pro in no time!

In this article, we successfully found that $g(2) = 1$ by carefully navigating the piecewise function definition. We also explored the real-world applications of these functions and worked through additional examples to solidify your understanding. So, go forth and conquer those piecewise functions!