Select The Correct Answer.What Is The Value Of $g(2)$?$g(x) = \left{\begin{array}{ll} \left(\frac{1}{2}\right)^{x-3}, & X\ \textless \ 2 \ X 3-9x 2+27x-25, & X \geq 2 \end{array}\right.$A. -1 B.

**Evaluating Piecewise Functions: A Step-by-Step Guide** =====================================================

Introduction

Piecewise functions are a type of mathematical function that consists of multiple sub-functions, each defined on a specific interval. These functions are commonly used in mathematics, physics, and engineering to model real-world phenomena. In this article, we will focus on evaluating piecewise functions, specifically the function $g(x) = \left{\begin{array}{ll} \left(\frac{1}{2}\right)^{x-3}, & x\ \textless \ 2 \ x^3-9x2+27x-25, & x \geq 2 \end{array}\right.$ and find the value of $g(2)$.

What is a Piecewise Function?

A piecewise function is a function that is defined by multiple sub-functions, each defined on a specific interval. The intervals are usually disjoint, meaning they do not overlap. The function is typically defined as:

$$f(x) = \left\{\begin{array}{ll} f_1(x), & x\ \textless \ a \\ f_2(x), & x \geq a \end{array}\right.$$

where $f_1(x)$ and $f_2(x)$ are the sub-functions, and $a$ is the point where the function changes.

Evaluating Piecewise Functions

To evaluate a piecewise function, we need to determine which sub-function to use based on the value of $x$. We do this by checking if $x$ is less than or greater than or equal to the point where the function changes.

Example: Evaluating $g(x)$

Let's evaluate the function $g(x) = \left{\begin{array}{ll} \left(\frac{1}{2}\right)^{x-3}, & x\ \textless \ 2 \ x^3-9x2+27x-25, & x \geq 2 \end{array}\right.$ at $x = 2$.

Since $x = 2$ is greater than or equal to $2$, we use the second sub-function, $x^3-9x2+27x-25$.

Step 1: Substitute $x = 2$ into the second sub-function

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

Step 2: Simplify the expression

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

Step 3: Evaluate the expression

$$g(2) = 1$$

Conclusion

In this article, we evaluated the piecewise function $g(x) = \left{\begin{array}{ll} \left(\frac{1}{2}\right)^{x-3}, & x\ \textless \ 2 \ x^3-9x2+27x-25, & x \geq 2 \end{array}\right.$ at $x = 2$. We determined that the second sub-function, $x^3-9x2+27x-25$, should be used since $x = 2$ is greater than or equal to $2$. We then $x = 2$ into the second sub-function, simplified the expression, and evaluated it to find that $g(2) = 1$.

Frequently Asked Questions

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by multiple sub-functions, each defined on a specific interval.

Q: How do I evaluate a piecewise function?

A: To evaluate a piecewise function, you need to determine which sub-function to use based on the value of $x$. You do this by checking if $x$ is less than or greater than or equal to the point where the function changes.

Q: What is the value of $g(2)$?

A: The value of $g(2)$ is $1$.

Q: Why do we need to use the second sub-function to evaluate $g(2)$?

A: We need to use the second sub-function because $x = 2$ is greater than or equal to $2$, which is the point where the function changes.

Q: Can I use the first sub-function to evaluate $g(2)$?

A: No, you cannot use the first sub-function to evaluate $g(2)$ because $x = 2$ is greater than or equal to $2$, which is not in the domain of the first sub-function.