Determining The Range Of Combined Functions F(x) And G(x)

In mathematics, understanding the behavior of functions is crucial, and one essential aspect of this is determining the range of a function. The range of a function represents the set of all possible output values (y-values) that the function can produce. When dealing with combinations of functions, such as the sum of two functions, it's important to analyze how these operations affect the overall range. This article delves into finding the range of a combined function, specifically $(f+g)(x)$, where $f(x) = |x| + 9$ and $g(x) = -6$. We will explore the properties of the absolute value function, constant functions, and how their combination impacts the resulting range. By carefully examining the individual functions and their interaction, we can accurately determine the bounds of the combined function's output values.

Defining the Functions

To begin, let's clearly define the functions involved. We are given two functions: $f(x) = |x| + 9$ and $g(x) = -6$. The function $f(x)$ is composed of two parts: the absolute value of $x$, denoted as $|x|$, and the constant 9. The absolute value function $|x|$ returns the non-negative magnitude of $x$, meaning it always outputs a value greater than or equal to zero. For example, $|3| = 3$ and $|-3| = 3$. This property is crucial for understanding the range of $f(x)$. The addition of 9 shifts the absolute value function upward by 9 units. On the other hand, the function $g(x) = -6$ is a constant function. This means that for any input $x$, the output is always -6. Constant functions have a very straightforward range, as they only produce a single value. Understanding these individual function behaviors is the first step in determining the range of their sum. Before we dive into the range of the combined function, let's take a closer look at the range of each function separately. The absolute value function $|x|$ has a range of $[0, \infty)$, meaning it can take any non-negative value. Adding 9 to this shifts the range upwards, so the range of $f(x) = |x| + 9$ is $[9, \infty)$. The constant function $g(x) = -6$ simply has a range of $\{-6\}$. With a solid understanding of these individual ranges, we can proceed to analyze the combined function $(f+g)(x)$.

Combining the Functions: (f+g)(x)

Now, let's combine the functions $f(x)$ and $g(x)$ to form the function $(f+g)(x)$. This is done by simply adding the two functions together: $(f+g)(x) = f(x) + g(x)$. Substituting the given functions, we have $(f+g)(x) = (|x| + 9) + (-6)$. Simplifying this expression, we get $(f+g)(x) = |x| + 3$. This new function, $(f+g)(x)$, is the sum of the absolute value of $x$ and the constant 3. To determine the range of $(f+g)(x)$, we need to consider how the absolute value function and the constant 3 interact. As we discussed earlier, the absolute value function $|x|$ always returns a non-negative value. Therefore, the smallest possible value of $|x|$ is 0, which occurs when $x = 0$. When $|x| = 0$, the value of $(f+g)(x)$ is $0 + 3 = 3$. This gives us a lower bound for the range of $(f+g)(x)$. Since $|x|$ can take any non-negative value, $|x| + 3$ can take any value greater than or equal to 3. As $x$ moves away from 0 in either the positive or negative direction, $|x|$ increases, and consequently, $(f+g)(x)$ also increases. There is no upper bound on the values that $|x|$ can take, so there is also no upper bound on the values of $(f+g)(x)$. This indicates that the range of $(f+g)(x)$ includes all values from 3 upwards to infinity. We can express this range mathematically using inequality notation and interval notation, which will help us in selecting the correct answer from the given options. Understanding how the individual function ranges combine is crucial in determining the final range of the combined function.

Determining the Range of (f+g)(x)

To accurately describe the range of $(f+g)(x) = |x| + 3$, we need to consider the possible output values of this function. As we established earlier, the absolute value function $|x|$ is always non-negative, meaning it will always be greater than or equal to 0. Consequently, the smallest possible value for $|x|$ is 0. When $x = 0$, we have $(f+g)(0) = |0| + 3 = 0 + 3 = 3$. This tells us that the minimum value of $(f+g)(x)$ is 3. Now, let's consider what happens as $x$ moves away from 0. Whether $x$ becomes a large positive number or a large negative number, the absolute value $|x|$ will increase. For example, if $x = 10$, then $|x| = 10$, and $(f+g)(10) = 10 + 3 = 13$. Similarly, if $x = -10$, then $|x| = |-10| = 10$, and $(f+g)(-10) = 10 + 3 = 13$. This shows that as $|x|$ increases, $(f+g)(x)$ also increases. Since $|x|$ can take any non-negative value, there is no upper limit to the values that $(f+g)(x)$ can take. Therefore, $(f+g)(x)$ can be any value greater than or equal to 3. We can express this range using inequality notation as $(f+g)(x) \geq 3$. This means that the function $(f+g)(x)$ will always output a value that is 3 or greater. There are no values less than 3 in the range of $(f+g)(x)$. Understanding this lower bound and the unbounded upper range is key to selecting the correct answer choice.

Analyzing the Answer Choices

Now that we have determined that the range of $(f+g)(x)$ is all values greater than or equal to 3, let's analyze the given answer choices to see which one correctly describes this range.

• A. $(f+g)(x) \geq 3$ for all values of $x$: This statement correctly describes the range we found. It says that the output of $(f+g)(x)$ is always greater than or equal to 3, which matches our analysis.
• B. $(f+g)(x) \leq 3$ for all values of $x$: This statement is incorrect. It suggests that the output of $(f+g)(x)$ is always less than or equal to 3, but we know that $(f+g)(x)$ can take values greater than 3.
• C. $(f+g)(x) \leq 6$ for all values of $x$: This statement is also incorrect. While $(f+g)(x)$ can take values less than or equal to 6, it's not limited to these values. $(f+g)(x)$ can be greater than 6 as well.
• D. $(f+g)(x) \geq 6$ for all values of $x$: This statement is incorrect because $(f+g)(x)$ is not always greater than or equal to 6. For example, when $x = 0$, $(f+g)(0) = 3$, which is not greater than or equal to 6.

By carefully comparing the range we determined with each answer choice, we can confidently identify the correct answer. It's crucial to not only understand the range itself but also how it is expressed in different mathematical notations. In this case, the inequality notation provides a clear and concise way to represent the range.

Conclusion and Key Takeaways

In conclusion, the range of the function $(f+g)(x)$, where $f(x) = |x| + 9$ and $g(x) = -6$, is accurately described by the inequality $(f+g)(x) \geq 3$ for all values of $x$. This means that the function's output will always be 3 or greater. The key to solving this problem lies in understanding the properties of the absolute value function and constant functions, as well as how their combination affects the resulting range. Remember that the absolute value function $|x|$ always returns a non-negative value, and a constant function always returns the same value regardless of the input. By combining these functions through addition, we shift the range accordingly. This problem highlights the importance of analyzing functions step-by-step, breaking them down into their component parts, and considering the impact of each part on the overall behavior of the function. When determining the range of a combined function, start by understanding the ranges of the individual functions and then consider how the operation (in this case, addition) affects the output values. This approach can be applied to various function combinations and will help you confidently solve similar problems in the future. Understanding function operations and range is a fundamental concept in mathematics, and mastering these skills will be beneficial for more advanced topics.