Function Reflection: Understanding The Range Of Y = (2/3) * 6^x

When we talk about the range of a function in mathematics, we're essentially discussing all the possible output values (the 'y' values) that the function can produce. It's like asking, "What numbers can this function possibly spit out?" Understanding this concept is crucial for graphing and analyzing mathematical relationships. For the function $f(x)=\frac{2}{3}(6)^x$, let's first break down what each part does. The term $(6)^x$ is an exponential function. Exponential functions, by their nature, have a base (in this case, 6) that is raised to a power (the variable 'x'). Without any other modifications, an exponential function like $(6)^x$ will always produce positive output values. For example, if x=1, $(6)^1=6$; if x=0, $(6)^0=1$; if x=-1, $(6)^{-1}=\frac{1}{6}$. Notice how all these results are greater than zero. The multiplication by $\frac{2}{3}$ in front of $(6)^x$ acts as a vertical stretch or compression. In this specific case, it's a vertical compression because the multiplier is between 0 and 1. However, this compression doesn't change the sign of the output. If $(6)^x$ is always positive, then $\frac{2}{3}(6)^x$ will also always be positive. Imagine stretching or squeezing a piece of string – its length might change, but if it started as a positive length, it will remain a positive length. Therefore, the original function $f(x)=\frac{2}{3}(6)^x$ has a range of all real numbers greater than 0. This means the 'y' values can be any positive number, no matter how small or how large, but they will never be zero or negative.

Now, let's dive into what happens when we reflect a function over the x-axis. Reflection is a geometric transformation that flips a graph across a line. When we reflect a graph over the x-axis, every point $(x, y)$ on the original graph becomes a point $(x, -y)$ on the reflected graph. Think of it like holding a mirror along the x-axis; the reflection is what you see on the other side. For our function $f(x)=\frac{2}{3}(6)^x$, this means that for every output 'y' that the original function produced, the reflected function will produce '-y'. If the original function's outputs were all positive numbers (greater than 0), then multiplying each of those positive numbers by -1 will result in all negative numbers (less than 0). For example, if the original function gave us an output of 2, the reflected function will give us -2. If the original gave us 1/6, the reflected will give us -1/6. This transformation directly impacts the range of the function. Since the original function $f(x)=\frac{2}{3}(6)^x$ had a range of all real numbers greater than 0 ($y > 0$), reflecting it across the x-axis will transform this range. The positive values will become negative. Therefore, the new range will be all real numbers less than 0 ($y < 0$). This is a fundamental concept in understanding how transformations affect the behavior and output possibilities of mathematical functions. It's not just about changing the formula; it's about understanding the geometric and algebraic consequences of those changes.

Let's solidify this understanding by considering the graphical interpretation of reflecting $f(x)=\frac{2}{3}(6)^x$ over the x-axis. The original function $f(x)=\frac{2}{3}(6)^x$ is an exponential growth function, but it's been compressed vertically. Because the base, 6, is greater than 1, the function increases rapidly as 'x' increases. As 'x' approaches negative infinity, $(6)^x$ approaches 0, so $f(x)$$ approaches 0 from above. This means the x-axis ($y=0$) is a horizontal asymptote for the original function. The graph hovers just above the x-axis for large negative values of 'x' and then shoots upwards as 'x' increases. The range of this original function, as we established, is $(0, \infty)$, or all positive real numbers. When we reflect this graph over the x-axis, everything that was above the x-axis gets flipped below it, and everything that was below gets flipped above. Since the original graph was entirely above the x-axis (except for the limiting behavior approaching $y=0$), the entire graph will now be below the x-axis. The horizontal asymptote, which was $y=0$, remains the same in terms of its value, but the graph will now approach it from below. So, for large negative values of 'x', the reflected function will approach 0 from below (meaning it will be slightly negative). As 'x' increases, the function will decrease rapidly into more and more negative values. The highest points the reflected graph can reach are values just below 0. Thus, the range of the reflected function is $(-\infty, 0)$, which translates to all real numbers less than 0. This graphical perspective confirms our algebraic conclusion about the range.

To further illustrate, let's consider the transformation of a single point. Pick a point on the original graph, say when $x=1$. Then $f(1) = \frac{2}{3}(6)^1 = \frac{2}{3} \times 6 = 4$. So, the point $(1, 4)$ is on the graph of $f(x)$. When we reflect this point over the x-axis, the new point becomes $(1, -4)$. This means that for the reflected function, let's call it $g(x)$, we have $g(1) = -4$. Notice that -4 is indeed a real number less than 0. Let's try another point. When $x=0$, $f(0) = \frac{2}{3}(6)^0 = \frac{2}{3} \times 1 = \frac{2}{3}$. The point is $(0, \frac{2}{3})$. Reflecting this over the x-axis gives us $(0, -\frac{2}{3})$. So, $g(0) = -\frac{2}{3}$, which is also less than 0. This systematic transformation of points clearly shows how the output values are inverted. If the original function could output any positive number, the reflected function can output any negative number. The magnitude of the output is preserved, but the sign is flipped. This is the essence of reflection over the x-axis. It's a direct inversion of the vertical values. Therefore, the range, which describes these vertical values, is also inverted in terms of its sign. From all real numbers greater than 0, it becomes all real numbers less than 0.

In conclusion, the function $f(x)=\frac{2}{3}(6)^x$ initially produces only positive real numbers as its output, meaning its range is $(0, \infty)$. When this function is reflected over the x-axis, every positive output value is transformed into its negative counterpart. Consequently, the range of the reflected function consists of all negative real numbers. This means the possible 'y' values for the transformed function are all numbers strictly less than zero. This aligns perfectly with option B, which states "all real numbers less than 0." Understanding function transformations, such as reflections, is a fundamental skill in mathematics that allows us to predict how changes in a function's definition will affect its graph and its output. For more on exponential functions and their transformations, you can explore resources like Khan Academy's section on exponential functions. They offer excellent explanations and practice problems that can further deepen your understanding of these concepts.