Finding The Range Of (f+g)(x) Where F(x)=|x|+9 And G(x)=-6

by ADMIN 59 views

Hey guys! Let's dive into a math problem that involves understanding function ranges. We're given two functions, f(x) = |x| + 9 and g(x) = -6, and our mission is to figure out the range of the combined function (f + g)(x). Don't worry, it's not as scary as it sounds! We'll break it down step by step.

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

Before we jump into the combined function, let's make sure we're crystal clear on what f(x) and g(x) are doing.

  • f(x) = |x| + 9: This function takes an input x, calculates its absolute value (|x|), and then adds 9. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. This is a crucial point! Because |x| is always greater than or equal to zero, the smallest value f(x) can be is when |x| is 0. In that case, f(x) = 0 + 9 = 9. For any other value of x, |x| will be a positive number, making f(x) greater than 9. So, f(x) will always be 9 or a number larger than 9. The key takeaway here is that the absolute value function ensures that the output is never negative, and adding 9 shifts the entire graph upwards by 9 units. This means the range of f(x) is all real numbers greater than or equal to 9. You can visualize this as a V-shaped graph with its vertex at the point (0, 9) and opening upwards. The two arms of the V extend infinitely upwards, indicating that the function can take on any value greater than or equal to 9. Think about it like this: no matter what number you plug in for x, the absolute value will always make it positive (or zero), and then adding 9 guarantees the result will be at least 9. This non-negative nature of the absolute value is fundamental to understanding the behavior of f(x) and, consequently, the behavior of (f + g)(x).

  • g(x) = -6: This is a constant function. No matter what input x we give it, the output is always -6. It's like a flat line stretching across the graph at y = -6. Constant functions are the simplest type of functions in terms of their range because there's only one possible output value. In this case, g(x) is always -6, making its range just the single value {-6}. There's no variation here; it's a straight, horizontal line. Understanding this constant behavior of g(x) is important when we combine it with f(x), as it will simply shift the range of f(x) by a fixed amount. Imagine a horizontal line drawn at -6 on the graph; that's g(x). It never changes, never deviates. This simplicity makes it easier to predict the outcome when we add it to another function.

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

Now comes the fun part: combining f(x) and g(x) to create a new function, (f + g)(x). This is a straightforward process. We simply add the expressions for f(x) and g(x) together:

(f + g)(x) = f(x) + g(x) = (|x| + 9) + (-6)

Simplifying this, we get:

(f + g)(x) = |x| + 3

So, (f + g)(x) takes an input x, calculates its absolute value, and then adds 3. Notice how similar this is to f(x)! The only difference is that instead of adding 9, we're adding 3. This shift will have a direct impact on the range of the function.

Let's think about the range of (f + g)(x) = |x| + 3 just like we did for f(x). The absolute value |x| is always greater than or equal to zero. Therefore, the smallest possible value for |x| is 0. When |x| = 0, (f + g)(x) = 0 + 3 = 3. This tells us that 3 is the minimum value that (f + g)(x) can take. For any non-zero value of x, |x| will be positive, and (f + g)(x) will be greater than 3. This is because adding a positive number to 3 will always result in a number larger than 3. The range of a function describes all the possible output values. In this case, since (f + g)(x) can be 3 and can be any number greater than 3, we can say that the range of (f + g)(x) is all real numbers greater than or equal to 3. To visualize this, imagine the V-shaped graph of |x| shifted upwards by 3 units. The vertex of the V is now at (0, 3), and the arms extend upwards, covering all y-values greater than or equal to 3. This graphical representation helps solidify the understanding of why the range is what it is.

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

Okay, we've done the heavy lifting. We know that (f + g)(x) = |x| + 3. We also know that the absolute value |x| is always non-negative (greater than or equal to 0). This means the smallest value |x| can be is 0.

So, the smallest value (f + g)(x) can be is when |x| = 0:

(f + g)(x) = |0| + 3 = 0 + 3 = 3

Since |x| will always be 0 or a positive number, (f + g)(x) will always be 3 or a number greater than 3. This means the range of (f + g)(x) is all values greater than or equal to 3.

The Answer

Now let's look at the answer choices. We're looking for a statement that accurately describes the range of (f + g)(x).

A. (f + g)(x) ≥ 3 for all values of x - This is exactly what we found! The function (f + g)(x) is always greater than or equal to 3.

B. (f + g)(x) ≤ 3 for all values of x - This is incorrect. We know (f + g)(x) is always greater than or equal to 3, not less than or equal to 3.

C. (f + g)(x) ≤ 6 for all values of x - This is also incorrect. While (f + g)(x) will be less than or equal to 6 for some values of x, it's not true for all values. For example, if x = 4, then (f + g)(4) = |4| + 3 = 7, which is greater than 6.

D. (f + g)(x) ≥ 6 for all values of x - This is incorrect as well. We know the minimum value of (f + g)(x) is 3, so it's not always greater than or equal to 6.

Therefore, the correct answer is A. (f + g)(x) ≥ 3 for all values of x.

Key Takeaways for Understanding Function Ranges

This problem is a great example of how understanding the properties of functions, especially the absolute value function, can help us determine their ranges. Here are some key takeaways:

  • Absolute Value: The absolute value function, |x|, always returns a non-negative value. This is a fundamental property to remember.
  • Constant Functions: Constant functions, like g(x) = -6, have a range consisting of a single value.
  • Combining Functions: When adding functions, the ranges are affected by the individual functions. In this case, adding the constant g(x) = -6 to f(x) = |x| + 9 shifted the range of f(x) downwards by 6 units.
  • Minimum Value: To find the range, it's often helpful to think about the minimum (or maximum) value the function can take.
  • Visualize: If possible, try to visualize the graphs of the functions. This can give you a better understanding of their behavior and ranges.

By understanding these concepts, you'll be well-equipped to tackle similar problems involving function ranges. Keep practicing, and you'll become a range-finding pro!

Remember, the key to mastering these kinds of problems is to break them down into smaller, manageable steps. First, understand the individual functions. Then, see how they combine. Finally, use your knowledge of the functions' properties to determine the range. You got this!