Range Of Exponential Function After Y-Axis Reflection

by Alex Johnson 54 views

Let's explore the range of the function f(x) = 2(1/4)^x after it undergoes a reflection over the y-axis. Understanding transformations of functions is crucial in mathematics, and this problem combines an exponential function with a reflection, offering a great opportunity to solidify these concepts.

Understanding the Original Function

Before diving into the reflection, let's first understand the basic behavior of the function f(x) = 2(1/4)^x. This is an exponential function with a base of 1/4, which is between 0 and 1. This means that as x increases, the function f(x) decreases, approaching 0 but never actually reaching it. The function is of the form f(x) = ab^x, where a = 2 and b = 1/4. The a value (2 in this case) represents a vertical stretch of the exponential function. Without the vertical stretch, that is, with a = 1, the range of b^x where 0 < b < 1 is (0, ∞). With the vertical stretch the range becomes (0, a∞) = (0, ∞). The domain of the exponential function is all real numbers. The horizontal asymptote of the exponential function is y = 0. The function is always positive because 2 is positive and (1/4)^x is always positive for any real number x. Therefore, the range of the original function f(x) = 2(1/4)^x is all real numbers greater than 0, or (0, ∞).

Key points about the original function:

  • It's an exponential decay function.
  • It approaches the x-axis (y=0) as x gets larger but never touches it.
  • It's always positive.
  • Its range is (0, ∞).

Exponential functions of the form f(x) = ab^x* are fundamental in describing phenomena like radioactive decay, population growth (or decline), and compound interest. Their behavior is dictated by the base b: if b > 1, it's exponential growth; if 0 < b < 1, it's exponential decay. The vertical stretch factor a simply scales the function, affecting its y-values but not fundamentally changing its increasing or decreasing nature. In our case, b = 1/4 signifies decay, and the function gets smaller as x increases. Grasping these basic principles is essential before dealing with transformations of exponential functions.

Reflecting Over the y-axis

Now, let's consider what happens when we reflect the function over the y-axis. Reflecting a function f(x) over the y-axis means replacing x with -x. So, our new function, g(x), becomes:

g(x) = f(-x) = 2(1/4)^{-x}

We can rewrite (1/4)^{-x} as (4)^x, so g(x) = 2(4)^x. This is now an exponential growth function, as the base is 4, which is greater than 1. However, the reflection over the y-axis does not affect the range of the function in this particular case. Why? Because the original function was already strictly positive. Replacing x by -x essentially mirrors the graph about the y-axis. The original function f(x) was decreasing as x increased, while the transformed function g(x) is increasing as x increases. The y values that the function takes are the same, but the x values that the function takes to get the y values are negated.

The y-axis reflection is a transformation that flips the graph horizontally. Each point (x, y) on the original graph is mapped to (-x, y) on the reflected graph. This transformation is vital in understanding function symmetry. If f(x) = f(-x), then the function is symmetric about the y-axis (an even function). If f(-x) = -f(x), then the function is symmetric about the origin (an odd function). Our original function, f(x) = 2(1/4)^x, is neither even nor odd. The reflection transforms it into g(x) = 2(4)^x, which is also neither even nor odd. What is important to consider is that the y-axis reflection only transforms the x value and not the y values of the original function. Since the y values are not transformed, the range will stay the same.

Determining the Range

Even though the reflection changes the function from decreasing to increasing, it doesn't change the set of possible y-values. The function g(x) = 2(4)^x will still approach 0 as x approaches negative infinity, and it will increase without bound as x increases. This means that the function will always be greater than 0, but it can take on any positive value. Therefore, the range of the reflected function is still all real numbers greater than 0. Thus, the range of the reflected function g(x) = 2(4)^x remains (0, ∞), which is all real numbers greater than 0.

Key points about the reflected function:

  • It's an exponential growth function.
  • It approaches the x-axis (y=0) as x gets smaller (more negative) but never touches it.
  • It's always positive.
  • Its range is (0, ∞).

Range determination often involves identifying the minimum and maximum possible output values of a function. For exponential functions, the range is heavily influenced by the horizontal asymptote and the presence of vertical shifts or reflections. In our case, the horizontal asymptote at y = 0 dictates that the function will never be zero or negative. The absence of vertical shifts means that the range starts immediately above the asymptote. Careful observation of the function's behavior as x approaches positive and negative infinity helps confirm the upper bound of the range.

Conclusion

The range of the function f(x) = 2(1/4)^x after it has been reflected over the y-axis is all real numbers greater than 0. The reflection over the y-axis transformed the function from an exponential decay to exponential growth, but it did not alter its range. This is because the y-axis reflection does not transform the y values of the original function. Therefore, the correct answer is:

A. all real numbers greater than 0

Understanding function transformations is a fundamental skill in mathematics. By understanding how reflections, stretches, and shifts affect a function, you can quickly analyze and understand the behavior of more complex functions. Always remember to consider the parent function and how each transformation alters its key characteristics, such as domain, range, intercepts, and asymptotes.

For further reading on exponential functions, you can visit Khan Academy's section on exponential functions.