Calculating G(x): A Step-by-Step Guide

Hey guys! Let's dive into a fun little math problem. We're gonna break down how to evaluate a piecewise function, specifically the function g(x). It's not as scary as it might look at first. We'll be going through the steps to find g(-5), g(-1), and g(2). Ready? Let's get started!

Understanding Piecewise Functions

Alright, so first things first: What exactly is a piecewise function? Well, basically, it's a function that's defined by different formulas or rules for different intervals of its input values (the x values). Think of it like this: depending on what x is, you use a specific equation. In our case, the function g(x) is defined as follows:

$g(x)=\left\{\begin{array}{ll} \frac{1}{4} x^2-5 & \text { if } x \neq-1 \\ -1 & \text { if } x=-1 \end{array}\right.$

See? There are two different rules here. The first rule, $\frac{1}{4} x^2 - 5$, applies to all x values except -1. The second rule, -1, only applies when x is exactly -1. This is super important because it tells us which equation we need to use when we're trying to find the value of the function at a specific x. It's like having different recipes for different ingredients. Knowing which recipe to use is key! Now, let's get into the nitty-gritty of how to find the values for the given inputs. The core concept to grasp is identifying the correct formula for each x value. The piecewise nature of the function means we must carefully observe the conditions attached to each part of the definition. Failing to do so can lead to incorrect results, so attention to detail is paramount. The function acts like a switchboard; based on the input, it routes the input through a particular formula. Thus, the objective becomes straightforward: accurately identify the appropriate equation based on the given x values. Once the appropriate equation has been selected, the calculation becomes a simple substitution and evaluation. This principle is not exclusive to this specific function; it applies to all piecewise functions, regardless of the complexity or the number of intervals.

Breaking Down g(x)

In our case, g(x) has two distinct behaviors. The first is a quadratic expression, $\frac{1}{4} x^2 - 5$, which is used for all x values except -1. The second is a constant function, g(x) = -1, specifically for the x value of -1. This distinction is crucial, as the value of g(x) when x = -1 is explicitly defined as -1, not the result of the quadratic equation. This type of definition prevents any potential ambiguity or conflict in the function's evaluation. It ensures a precise and well-defined output for all inputs. The importance of understanding these nuances cannot be overstated, as they form the foundation for correctly evaluating any piecewise function. Moreover, these concepts extend beyond mere calculations; they deepen one's understanding of mathematical functions and their properties. The ability to correctly interpret and evaluate a piecewise function is a fundamental skill in mathematics, useful not only in academic settings but also in various real-world applications. By carefully analyzing the different segments of the piecewise function, one can predict and compute function values accurately, given any input value within the function's domain.

Finding g(-5)

Alright, let's find g(-5). We need to figure out which part of the function definition to use. Notice that -5 is not equal to -1. Therefore, we're going to use the first rule: $\frac{1}{4} x^2 - 5$. Now, we simply substitute -5 for x in that equation:

g(-5) = $\frac{1}{4} (-5)^2 - 5$

Let's break this down step-by-step:

• $(-5)^2 = 25$ (because a negative times a negative is a positive, duh!)
• $\frac{1}{4} * 25 = 6.25$
• $6.25 - 5 = 1.25$

So, g(-5) = 1.25. Easy peasy, right?

Step-by-step calculation for g(-5)

The process of computing g(-5) involves a series of clear, straightforward steps. First, we identify that the input value, x = -5, does not satisfy the condition x = -1. Consequently, we utilize the first rule of the function, which is $\frac{1}{4} x^2 - 5$. In this step, we substitute -5 into the equation: $g(-5) = \frac{1}{4} (-5)^2 - 5$. This leads us to the next step, where we must evaluate the square of -5. Remember that when multiplying a negative number by itself, the result is positive. Therefore, $(-5)^2$ is equal to 25. The equation then becomes $g(-5) = \frac{1}{4} * 25 - 5$. In the following step, the fraction $\frac{1}{4}$ is multiplied by 25, which results in 6.25. Our updated equation is $g(-5) = 6.25 - 5$. Finally, we subtract 5 from 6.25 to obtain the final result, which is 1.25. Therefore, $g(-5) = 1.25$. This methodical approach highlights the importance of following the order of operations and accurately substituting the given values into the function's formula. Each step is crucial, and any error can lead to an incorrect result. This is a common method for handling this type of function, which helps ensure that we reach an accurate solution. The ability to perform such computations is a fundamental skill in mathematics, as it provides a foundation for more advanced concepts and applications.

Finding g(-1)

Now, let's find g(-1). This is where it gets super simple. Look at the function definition again. See the part that says "if x = -1"? That's the rule we use here. It tells us that g(-1) = -1. Bam! Done.

Direct Application of the Function Definition for g(-1)

Finding g(-1) is a direct application of the piecewise function's definition. We observe that the input value is -1, which precisely matches one of the function's conditions. According to the definition, when x is equal to -1, g(x) is directly defined as -1. This means we bypass the quadratic equation altogether and go directly to the given value. The advantage of this type of function definition is its clarity and precision. There is no potential for misinterpretation or ambiguity. The explicit definition for x = -1 simplifies the calculation and removes any need for further computation. The result, g(-1) = -1, immediately follows from the function's specification. This aspect of the function design ensures that it handles the specific input precisely as intended. It showcases how different parts of a piecewise function work in tandem to produce the final output. The direct definition for g(-1) is also a reminder that the function provides specific outputs based on the input values, and sometimes the answer is provided directly, with no complex calculations needed.

Finding g(2)

Alright, last one! Let's find g(2). Since 2 is not equal to -1, we're going to use the first rule again: $\frac{1}{4} x^2 - 5$. Let's plug in 2 for x:

g(2) = $\frac{1}{4} (2)^2 - 5$

Let's solve it:

• $(2)^2 = 4$
• $\frac{1}{4} * 4 = 1$
• $1 - 5 = -4$

So, g(2) = -4. Awesome!

Step-by-Step Calculation for g(2)

The process for calculating g(2) parallels the process for g(-5). Firstly, we identify that the input value, x = 2, does not satisfy the condition x = -1. Consequently, we utilize the first rule of the function, $\frac{1}{4} x^2 - 5$. We then substitute 2 into this equation, yielding $g(2) = \frac{1}{4} (2)^2 - 5$. The next step involves evaluating the square of 2, which equals 4. The equation then becomes $g(2) = \frac{1}{4} * 4 - 5$. Subsequently, we multiply the fraction $\frac{1}{4}$ by 4, resulting in 1. Our updated equation is $g(2) = 1 - 5$. Finally, we subtract 5 from 1, to derive the final answer of -4. Hence, g(2) = -4. This stepwise approach reinforces the importance of the correct substitution and adherence to the order of operations. It is also an example of how the function reacts to various inputs. It is crucial to remember that each step must be performed with precision to avoid any computational errors. This method demonstrates how straightforward the calculation process is, provided the rules are followed exactly as stated. The ability to perform such calculations is key to success in various mathematical contexts.

Conclusion

So there you have it, guys! We successfully found the values of g(-5), g(-1), and g(2) by carefully applying the rules of our piecewise function. Remember to always pay attention to the conditions for each part of the function. Keep practicing, and you'll be a piecewise function pro in no time! Peace out!

Summary of Results

To recap our findings:

• g(-5) = 1.25 (Using the rule $\frac{1}{4} x^2 - 5$)
• g(-1) = -1 (Using the rule g(x) = -1 when x = -1)
• g(2) = -4 (Using the rule $\frac{1}{4} x^2 - 5$)

These results clearly illustrate how the function g(x) behaves differently based on the input value. The ability to evaluate such functions is a foundational skill in mathematics, enabling further exploration of more complex concepts. Through careful attention to detail and a systematic approach, we are able to easily determine the values of these functions given their specific inputs. The contrast in the calculation methods for g(-1) compared to the other two values also highlights the importance of understanding the complete function definition. This kind of problem showcases why understanding the conditions attached to the different parts of a function is very important to avoid any confusion or mistake.