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

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Hey guys! Today, we're diving into the fascinating world of functions, specifically focusing on how to determine the range of a combined function. We're tackling a problem where we have two functions, f(x) = |x| + 9 and g(x) = -6, and our mission is to figure out the range of their sum, (f + g)(x). Sounds like a fun challenge, right? Let's break it down step by step so we can all understand it perfectly.

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 individually. The function f(x) = |x| + 9 is a classic example of an absolute value function with a little twist. Remember, the absolute value of a number, denoted by |x|, is its distance from zero. So, |x| is always non-negative; it's either zero or a positive number. Now, we're adding 9 to this absolute value. This means the smallest possible value of f(x) occurs when |x| is zero, which happens when x is zero. So, f(0) = |0| + 9 = 9. For any other value of x, |x| will be positive, making f(x) greater than 9. This tells us that the range of f(x) is all real numbers greater than or equal to 9. We can visualize this as a V-shaped graph that opens upwards, with its lowest point at the y-value of 9.

Now, let's talk about g(x) = -6. This function is much simpler. No matter what value of x we plug in, g(x) is always -6. It's a constant function. Think of it as a horizontal line on the graph that intersects the y-axis at -6. The range of g(x) is just the single value -6. Understanding these individual functions is crucial because it sets the stage for understanding how they behave when combined.

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

Okay, now for the exciting part: combining the functions! When we talk about (f + g)(x), we simply mean adding the outputs of the two functions for the same input x. Mathematically, this looks like (f + g)(x) = f(x) + g(x). So, to find (f + g)(x), we take the expression for f(x), which is |x| + 9, and add it to the expression for g(x), which is -6. Let's do that:

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

Simplifying this expression, we get:

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

This new function, |x| + 3, is what we're really interested in. It tells us how the combined function behaves. Notice that it's still an absolute value function, but this time, we're adding 3 instead of 9. This vertical shift is key to determining the range of (f + g)(x). Because |x| is always non-negative, the smallest value it can be is 0. This occurs when x is 0. So, the smallest value of (f + g)(x) is |0| + 3 = 3. For any other value of x, |x| will be greater than 0, making (f + g)(x) greater than 3.

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

Alright, we've arrived at the crucial question: What's the range of (f + g)(x) = |x| + 3? We've already figured out that the smallest possible value of (f + g)(x) is 3, which happens when x is 0. As x moves away from 0 in either direction (positive or negative), |x| increases, and so does (f + g)(x). There's no upper limit to how large |x| can get, so there's also no upper limit to how large (f + g)(x) can get. This means (f + g)(x) can take on any value greater than or equal to 3. In mathematical terms, we say the range of (f + g)(x) is all real numbers greater than or equal to 3.

So, let's put it all together. We started with f(x) = |x| + 9 and g(x) = -6. We combined them to get (f + g)(x) = |x| + 3. We then analyzed this combined function and determined that its range is all real numbers greater than or equal to 3. This means that (f + g)(x) ≥ 3 for all values of x. Understanding how the absolute value function behaves and how vertical shifts affect the range was key to solving this problem. You guys nailed it!

Conclusion

In conclusion, the range of (f + g)(x) is all values greater than or equal to 3. Remember, breaking down the problem into smaller steps—understanding the individual functions, combining them, and then analyzing the result—is a powerful strategy for tackling these types of questions. Keep practicing, and you'll become function masters in no time! You got this!

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