Y-Intercept Of A Piecewise Function: A Detailed Guide

$g(x)=\left\{\begin{array}{ll} x-4 ; & x \leq 1 \\ -3 ; & 1 < x \leq 3 \\ x-6 ; & x>3 \end{array}\right.$

Which statement is true? A. The $y$-intercept of $g$ cannot be determined from the given information. B. The $y$-intercept of $g$ is -4. C. The $y$-intercept of $g$ is -6. D. The $y$-intercept of $g$ is 0.

Understanding Piecewise Functions and Y-Intercepts

Let's dive into the world of piecewise functions and how to find their y-intercepts. A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. In simpler terms, it's like having different rules for different parts of the x-axis. Finding the y-intercept of such a function requires a bit of care, as we need to determine which piece of the function applies when x = 0. The y-intercept is the point where the graph of the function intersects the y-axis. This occurs when the value of $x$ is equal to 0. To find it, we simply substitute $x = 0$ into the appropriate part of the piecewise function and solve for $g(0)$. Piecewise functions are incredibly useful for modeling real-world situations where different conditions lead to different outcomes. For instance, think about how shipping costs might vary based on the weight of a package or how income tax brackets work. Each "piece" of the function dictates the output for a specific range of inputs, making them versatile tools in mathematics and various applications. Knowing how to work with these functions, especially how to find key features like the y-intercept, is crucial for understanding their behavior and using them effectively. Understanding these nuances is the key to correctly determining the y-intercept and making informed decisions based on the function's behavior. Remember, a thorough understanding of the function's structure is your best tool.

Analyzing the Given Piecewise Function

In our case, we have the piecewise function g(x) defined as follows:

$g(x)=\left\{\begin{array}{ll} x-4 ; & x \leq 1 \\ -3 ; & 1 < x \leq 3 \\ x-6 ; & x>3 \end{array}\right.$

This function has three different rules depending on the value of x. The first rule, g(x) = x - 4, applies when x is less than or equal to 1. The second rule, g(x) = -3, applies when x is greater than 1 but less than or equal to 3. The third rule, g(x) = x - 6, applies when x is greater than 3. To find the y-intercept, we need to evaluate g(0). Since 0 is less than or equal to 1, we use the first rule: g(x) = x - 4. Substituting x = 0 into this rule gives us g(0) = 0 - 4 = -4. Therefore, the y-intercept of the function g(x) is -4. Let's break down why we chose the first rule. The y-intercept occurs where x=0. Looking at the conditions for each piece of the function, only the first one, x ≤ 1, includes x=0. Therefore, we must use the corresponding function, g(x) = x - 4, to calculate the y-intercept. This systematic approach ensures that we select the correct piece of the function for the given x-value, leading to an accurate determination of the y-intercept. Always remember to carefully examine the domain conditions of each piece to avoid errors. Grasping this concept is vital for correctly interpreting and applying piecewise functions in various mathematical and real-world scenarios. So the important point to remember is to find the interval where x=0 belongs to, and apply the corresponding equation of that interval.

Determining the Correct Statement

Now that we've found the y-intercept, let's evaluate the given statements:

A. The $y$-intercept of $g$ cannot be determined from the given information. B. The $y$-intercept of $g$ is -4. C. The $y$-intercept of $g$ is -6. D. The $y$-intercept of $g$ is 0.

Based on our analysis, we found that the y-intercept is -4. Therefore, statement B is the correct statement. The other statements are incorrect because we were able to determine the y-intercept, and it is not -6 or 0. Specifically, statement A is incorrect because we successfully determined the y-intercept by evaluating the function at x = 0. Statement C is incorrect because it uses the wrong piece of the function; g(x) = x - 6 applies only when x > 3. Statement D is also incorrect, as our calculation clearly showed that g(0) = -4. Understanding why each statement is either correct or incorrect reinforces the importance of carefully analyzing the piecewise function and applying the appropriate rule based on the value of x. This process ensures that you arrive at the correct conclusion and avoid common mistakes when working with piecewise functions. By systematically eliminating incorrect options, you can confidently identify the correct statement and demonstrate a solid understanding of the concept.

Conclusion

In conclusion, when given a piecewise function and asked to find its y-intercept, the key is to identify which piece of the function applies when $x = 0$. In this case, the correct answer is B. The $y$-intercept of $g$ is -4. Always carefully examine the conditions for each piece of the function to ensure you are using the correct rule. Understanding piecewise functions is fundamental in many areas of mathematics and its applications, and mastering the process of finding key features like the y-intercept is essential for problem-solving. Remember to always check which interval x=0 belongs to, and then apply that interval to the relevant equation. This will always give you the correct y intercept value. For further exploration of piecewise functions, you can visit Khan Academy's section on piecewise functions.