Determining The Range Of Composite Functions F(x) = |x| + 9 And G(x) = -6

In this article, we will delve into the process of determining the range of a composite function, specifically $(f+g)(x)$, given the individual functions $f(x) = |x| + 9$ and $g(x) = -6$. Understanding the range of a function is crucial in mathematics as it defines the set of all possible output values. We will explore how to combine these functions, analyze the resulting expression, and identify the minimum and maximum values to accurately describe the range.

Defining the Functions f(x) and g(x)

Before we begin, let's clearly define the functions we will be working with. The function $f(x)$ is defined as the absolute value of $x$ plus 9. The absolute value function, denoted by $|x|$, returns the non-negative magnitude of $x$, meaning it outputs the distance of $x$ from zero. Thus, $|x|$ is always greater than or equal to zero. Adding 9 to this value shifts the entire function upwards by 9 units. Therefore, $f(x) = |x| + 9$ will always produce values greater than or equal to 9. This is a crucial understanding because it sets the foundation for determining the range of the composite function.

On the other hand, the function $g(x)$ is a constant function defined as -6. This means that for any input value of $x$, the output will always be -6. Constant functions are straightforward to deal with as they do not vary with the input and have a range consisting of only a single value. The simplicity of $g(x)$ allows us to focus primarily on the behavior of $f(x)$ when we combine them.

Understanding these individual functions and their properties is paramount to determining the range of their combination. $f(x)$ introduces the variability due to the absolute value, while $g(x)$ provides a constant offset. Now that we have clearly defined $f(x)$ and $g(x)$, we can move on to combining them and finding the range of the resulting function.

Combining f(x) and g(x) to Form (f+g)(x)

To find $(f+g)(x)$, we simply add the two functions together. This is a fundamental operation in function algebra, where we combine functions by performing arithmetic operations on their expressions. In this case, we add the expression for $f(x)$ to the expression for $g(x)$.

Mathematically, this is represented as: $(f+g)(x) = f(x) + g(x)$. Substituting the given functions, we have $(f+g)(x) = (|x| + 9) + (-6)$. Simplifying this expression, we combine the constant terms 9 and -6, resulting in $(f+g)(x) = |x| + 3$. This new function, $(f+g)(x)$, represents the sum of the absolute value of $x$ and 3. This simplification is a crucial step because it allows us to analyze the range of the combined function more easily.

The function $(f+g)(x) = |x| + 3$ is a transformation of the absolute value function. The addition of 3 shifts the standard absolute value function upwards by 3 units. This shift directly affects the range of the function. The absolute value of $x$, $|x|$, is always non-negative, meaning it can be zero or any positive number. Therefore, the smallest value that $|x|$ can take is 0. Adding 3 to this smallest value gives us the minimum value of the function $(f+g)(x)$.

Understanding this combination and simplification process is essential for determining the range. By adding the functions and simplifying, we have obtained a clear expression for $(f+g)(x)$ that allows us to easily identify its minimum value and subsequently its range. Now that we have the simplified form, we can proceed to analyze its range.

Analyzing the Range of (f+g)(x) = |x| + 3

Now that we have the simplified function $(f+g)(x) = |x| + 3$, let's delve into determining its range. The range of a function is the set of all possible output values. To find the range, we need to identify the minimum and maximum values that the function can attain.

As we previously discussed, the absolute value function $|x|$ is always non-negative. This means that the smallest possible value for $|x|$ is 0, which occurs when $x = 0$. When $|x| = 0$, the function $(f+g)(x) = |x| + 3$ becomes $(f+g)(0) = 0 + 3 = 3$. This tells us that the minimum value of the function $(f+g)(x)$ is 3. Since $|x|$ can only be 0 or positive, $(f+g)(x)$ will never be less than 3.

On the other hand, as $x$ moves away from 0 in either the positive or negative direction, the value of $|x|$ increases without bound. Consequently, $(f+g)(x) = |x| + 3$ also increases without bound. This means there is no maximum value for the function. As $x$ approaches infinity or negative infinity, $(f+g)(x)$ also approaches infinity.

Therefore, the range of $(f+g)(x)$ consists of all values greater than or equal to 3. We can express this range using inequality notation as $(f+g)(x) ext{ ≥ } 3$. This notation clearly indicates that the function's output values start at 3 and extend upwards indefinitely. Understanding how the absolute value function affects the range is critical in this analysis. The non-negativity of $|x|$ directly contributes to the minimum value of the composite function.

Determining the Correct Option

Having analyzed the range of $(f+g)(x)$, we can now determine the correct option from the given choices. We have established that the range of $(f+g)(x) = |x| + 3$ is all values greater than or equal to 3, which is mathematically represented as $(f+g)(x) ext{ ≥ } 3$.

Let's examine the options:

A. $(f+g)(x) ext{ ≥ } 3$ for all values of $x$ B. $(f+g)(x) ext{ ≤ } 3$ for all values of $x$ C. $(f+g)(x) ext{ ≤ } 6$ for all values of $x$ D. $(f+g)(x) ext{ ≥ } 6$ for all values of $x$

Comparing our derived range with the options, it is clear that option A, $(f+g)(x) ext{ ≥ } 3$ for all values of $x$, accurately describes the range of the function. This option aligns perfectly with our analysis, which showed that the minimum value of $(f+g)(x)$ is 3 and that the function can take any value greater than 3.

Options B, C, and D are incorrect. Option B states that $(f+g)(x)$ is less than or equal to 3, which contradicts our finding that 3 is the minimum value. Options C and D propose incorrect upper and lower bounds for the range, respectively. Therefore, by carefully analyzing the function and its range, we can confidently identify option A as the correct answer. Selecting the right answer demonstrates a comprehensive understanding of function composition and range determination.

Conclusion

In conclusion, by combining the functions $f(x) = |x| + 9$ and $g(x) = -6$, we obtained the composite function $(f+g)(x) = |x| + 3$. Analyzing this function, we determined that its range is all values greater than or equal to 3. This conclusion was reached by understanding the properties of the absolute value function and how adding a constant shifts the range. Therefore, the correct answer is A. $(f+g)(x) ext{ ≥ } 3$ for all values of $x$.

This exercise demonstrates the importance of understanding function composition, range analysis, and the behavior of absolute value functions. By carefully breaking down the problem and analyzing each component, we were able to accurately determine the range of the composite function. This approach is applicable to a wide range of mathematical problems involving functions and their properties. Mastering these concepts provides a solid foundation for more advanced mathematical studies and applications.