Comparing Ranges: G(x) = 6/x Vs F(x) = 1/x

Oct 27, 2025 by ADMIN 43 views

Let's dive into comparing the ranges of two functions: g(x) = 6/x and its parent function f(x) = 1/x. Understanding how functions transform and affect their ranges is a crucial concept in mathematics. We'll break down each function, analyze their ranges, and see how they stack up against each other. So, grab your thinking caps, and let's get started!

Understanding the Parent Function: f(x) = 1/x

First off, let's get cozy with the parent function, f(x) = 1/x. This function is a classic example of a reciprocal function, and it forms the foundation for understanding g(x) = 6/x. When we talk about the range of a function, we're essentially asking: "What are all the possible y-values (output values) that this function can produce?" Think about plugging in different values for x in f(x) = 1/x.

If x is a large positive number, like 1000, then f(x) becomes 1/1000, which is very close to zero.
If x is a small positive number, like 0.001, then f(x) becomes 1/0.001 = 1000, a large positive number.
If x is a large negative number, like -1000, then f(x) becomes 1/-1000, which is a small negative number (close to zero).
If x is a small negative number, like -0.001, then f(x) becomes 1/-0.001 = -1000, a large negative number.

Notice anything interesting? As x gets closer and closer to zero (from either the positive or negative side), the value of f(x) shoots off towards positive or negative infinity. However, there's one value that x cannot be: zero itself. If we try to plug in x = 0, we get f(0) = 1/0, which is undefined. This means there's a vertical asymptote at x = 0.

Also, notice that f(x) can get incredibly close to zero, but it will never actually equal zero. No matter what non-zero value we plug in for x, 1/x will never be exactly zero. This means there's a horizontal asymptote at y = 0.

So, what's the range of f(x) = 1/x? It's all real numbers except zero. We can write this mathematically as:

Range of f(x): {y | y ∈ ℝ, y ≠ 0}

Where ℝ represents the set of all real numbers.

Analyzing the Transformed Function: g(x) = 6/x

Now, let's turn our attention to g(x) = 6/x. This function looks pretty similar to f(x) = 1/x, but there's a key difference: the 6 in the numerator. This 6 acts as a vertical stretch. It's like taking the graph of f(x) = 1/x and pulling it away from the x-axis. But what does this do to the range?

Think about it this way: we're essentially multiplying all the output values of f(x) by 6 to get g(x). So, if f(x) could produce any real number except zero, multiplying those values by 6 will still result in any real number except zero. The vertical stretch doesn't introduce any new restrictions on the possible output values.

Let's consider the same scenarios we used for f(x):

If x is a large positive number, like 1000, then g(x) becomes 6/1000, which is close to zero.
If x is a small positive number, like 0.001, then g(x) becomes 6/0.001 = 6000, a large positive number.
If x is a large negative number, like -1000, then g(x) becomes 6/-1000, which is a small negative number (close to zero).
If x is a small negative number, like -0.001, then g(x) becomes 6/-0.001 = -6000, a large negative number.

Just like with f(x), x cannot be zero in g(x) because division by zero is undefined. And, g(x) can get arbitrarily close to zero, but it will never actually equal zero. The vertical stretch affects how quickly the function approaches infinity or zero, but it doesn't change the fundamental fact that g(x) can never be zero.

Therefore, the range of g(x) = 6/x is also all real numbers except zero. We can write this mathematically as:

Range of g(x): {y | y ∈ ℝ, y ≠ 0}

Comparing the Ranges

So, let's bring it all together. We've established that:

The range of f(x) = 1/x is all real numbers except zero.
The range of g(x) = 6/x is all real numbers except zero.

This means that the ranges of the two functions are the same! The vertical stretch caused by the factor of 6 in g(x) doesn't change the set of possible output values. It only affects how those values are distributed.

To really solidify this, think about the graphs of the two functions. Both graphs will have vertical asymptotes at x = 0 and horizontal asymptotes at y = 0. The graph of g(x) will simply be stretched vertically compared to the graph of f(x), but it will still cover the same range of y-values.

Key Takeaways

The range of a function is the set of all possible output values (y-values).
The parent function f(x) = 1/x has a range of all real numbers except zero.
The transformed function g(x) = 6/x also has a range of all real numbers except zero.
A vertical stretch (like multiplying the function by a constant) can change the shape of the graph, but it doesn't necessarily change the range.

Visualizing the Functions

To further illustrate this, let's think about what the graphs of these functions look like. Both f(x) = 1/x and g(x) = 6/x are hyperbolas. Hyperbolas have a characteristic shape with two separate branches that approach asymptotes.

For both functions:

There's a vertical asymptote at x = 0. This means the functions get closer and closer to the vertical line x = 0 but never actually touch it.
There's a horizontal asymptote at y = 0. This means the functions get closer and closer to the horizontal line y = 0 but never actually touch it.

The key difference lies in the vertical stretch. The graph of g(x) = 6/x will appear "stretched" vertically compared to f(x) = 1/x. Imagine grabbing the graph of f(x) and pulling it upwards and downwards away from the x-axis. That's essentially what the multiplication by 6 does.

However, even with this stretch, the graph of g(x) still covers all y-values except zero. It just reaches those values more quickly as you move away from the vertical asymptote.

Range and Transformations: A Broader View

This example highlights an important principle about function transformations and their effect on the range. While certain transformations, like vertical stretches and compressions, can change the appearance of a graph, they don't always alter the range.

Here are some other transformations to consider:

Vertical Shifts: Adding or subtracting a constant outside the function (e.g., f(x) + 2) will shift the graph up or down, and this will change the range. For instance, if we had a function h(x) = 1/x + 2, its range would be all real numbers except 2.
Horizontal Shifts: Adding or subtracting a constant inside the function (e.g., f(x + 2)) will shift the graph left or right. This doesn't affect the range.
Reflections: Reflecting the graph across the x-axis (multiplying the function by -1) will change the sign of the y-values, and this can affect the range if the original range wasn't symmetric about zero.

Understanding how these transformations impact the range is crucial for analyzing functions and their behavior. Always consider the parent function, the type of transformation, and how it might affect the set of possible output values.

Common Mistakes to Avoid

When comparing the ranges of functions, it's easy to fall into some common traps. Here are a few things to watch out for:

Confusing Range with Domain: The domain is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Make sure you're focusing on the y-values when determining the range.
Ignoring Asymptotes: Asymptotes play a crucial role in defining the range of many functions, especially rational functions like f(x) = 1/x and g(x) = 6/x. Remember that the function can get arbitrarily close to an asymptote but never actually reach it.
Overgeneralizing Transformations: Don't assume that all transformations will affect the range in the same way. As we discussed, vertical stretches don't necessarily change the range, while vertical shifts do.
Not Considering Special Cases: Always think about any special cases or restrictions on the function. For example, division by zero is undefined, so you need to exclude any x-values that would lead to division by zero.

By being mindful of these potential pitfalls, you can improve your accuracy and confidence when analyzing the ranges of functions.

Practice Problems

To really master this concept, let's try a few practice problems:

What is the range of the function h(x) = -1/x?
How does the range of k(x) = 1/(x - 2) compare to the range of f(x) = 1/x?
What is the range of m(x) = 3 + 1/x?

Work through these problems, and be sure to think about the parent function, any transformations, and the presence of asymptotes. The more you practice, the better you'll become at determining the ranges of various functions.

Conclusion

In conclusion, comparing the ranges of g(x) = 6/x and its parent function f(x) = 1/x reveals a fundamental concept in function transformations. While g(x) undergoes a vertical stretch, its range remains the same as f(x): all real numbers except zero. This is because the vertical stretch doesn't introduce any new restrictions on the possible output values. Understanding how transformations affect the range is crucial for analyzing functions and their behavior. So keep practicing, keep exploring, and you'll become a range-finding pro in no time! Remember guys, math is all about understanding the underlying principles and applying them in different contexts. Keep up the great work!