Exploring The Properties And Graph Of The Exponential Function Y=(1/3)^x
Which of the following statements are true about the graph of the function $y=\left(\frac{1}{3}\right)^x$?
In the realm of mathematical functions, exponential functions hold a prominent position, playing a pivotal role in modeling various phenomena across diverse fields, from population growth and radioactive decay to compound interest and financial investments. Among the vast family of exponential functions, the function y = (1/3)^x stands out as a classic example of exponential decay, exhibiting a unique set of properties that make it a fascinating subject of study. In this comprehensive article, we will delve deep into the intricacies of this function, unraveling its key characteristics and exploring its graphical representation. Our primary focus will be on examining several statements about the graph of y = (1/3)^x, meticulously evaluating their validity and providing a clear understanding of why they hold true or false. By the end of this exploration, you will have a solid grasp of the behavior of exponential decay functions and their graphical manifestations.
Before we embark on our journey of analyzing the statements about the graph of y = (1/3)^x, let's first establish a firm foundation by dissecting the function itself. This function belongs to the broader family of exponential functions, which are generally expressed in the form y = a^x, where a is a positive constant known as the base. The base a plays a crucial role in determining the behavior of the function. When a is greater than 1, the function represents exponential growth, characterized by a rapid increase in y as x increases. Conversely, when a is between 0 and 1, as is the case with our function y = (1/3)^x, we encounter exponential decay. In exponential decay, the value of y decreases as x increases, approaching zero as x tends towards infinity. This fundamental understanding of the base's influence sets the stage for our investigation into the specific properties of y = (1/3)^x.
Our first statement asserts that the function y = (1/3)^x is increasing. To determine the veracity of this statement, we must delve into the concept of increasing functions. A function is deemed increasing if its value consistently rises as the input x increases. In other words, for any two values of x, say x1 and x2, where x1 < x2, the function is increasing if f(x1) < f(x2). Now, let's consider our function y = (1/3)^x. As x increases, the exponent becomes larger, causing the base (1/3) to be raised to a higher power. Since the base is a fraction between 0 and 1, raising it to a higher power results in a smaller value. For instance, (1/3)^2 = 1/9, which is less than (1/3)^1 = 1/3. This pattern holds true for all values of x, indicating that as x increases, y decreases. Therefore, the statement that the function is increasing is false. In fact, as we discussed earlier, this function exhibits exponential decay, meaning it is a decreasing function.
Having debunked the notion that the function is increasing, let's turn our attention to the second statement, which posits that the function y = (1/3)^x is decreasing. As we alluded to in our discussion of the first statement, a function is considered decreasing if its value consistently declines as the input x increases. Mathematically, this translates to f(x1) > f(x2) for any two values of x, x1 and x2, where x1 < x2. Our exploration of the function's behavior in the previous section provides ample evidence to support this statement. We observed that as x grows larger, the value of y = (1/3)^x diminishes. This is a hallmark of exponential decay, where the function's output progressively decreases as the input increases. To further solidify this understanding, consider the following examples:
- When x = -1, y = (1/3)^(-1) = 3
- When x = 0, y = (1/3)^(0) = 1
- When x = 1, y = (1/3)^(1) = 1/3
- When x = 2, y = (1/3)^(2) = 1/9
As we can clearly see, as x increases, the value of y decreases. This consistent decline in the function's value confirms that the function y = (1/3)^x is indeed decreasing. Therefore, the statement that the function is decreasing is true.
Now, let's investigate the third statement, which claims that the x-intercept of the graph of y = (1/3)^x is (1, 0). To determine the x-intercept, we need to find the point where the graph intersects the x-axis. This occurs when the value of y is equal to zero. So, we need to solve the equation (1/3)^x = 0. However, a crucial characteristic of exponential functions comes into play here. Exponential functions, regardless of their base, never actually reach zero. As x approaches positive infinity, the value of (1/3)^x gets infinitesimally close to zero, but it never truly touches the x-axis. This is because any positive number raised to any power will always be greater than zero. Consequently, the graph of y = (1/3)^x has no x-intercept. The curve gets closer and closer to the x-axis but never intersects it. Therefore, the statement that the x-intercept is (1, 0) is false. The graph of y = (1/3)^x does not have an x-intercept.
Moving on to the fourth statement, we are presented with the assertion that the y-intercept of the graph of y = (1/3)^x is (0, 1). The y-intercept is the point where the graph intersects the y-axis, which occurs when the value of x is equal to zero. To find the y-intercept, we simply substitute x = 0 into the function: y = (1/3)^0. Any non-zero number raised to the power of zero is equal to 1. Therefore, y = (1/3)^0 = 1. This means that when x is 0, y is 1, and the graph intersects the y-axis at the point (0, 1). This aligns perfectly with the statement, confirming its validity. Therefore, the statement that the y-intercept is (0, 1) is true. This is a characteristic feature of exponential functions of the form y = a^x; they always have a y-intercept at (0, 1), regardless of the value of the base a.
Finally, let's tackle the fifth statement, which concerns the range of the function y = (1/3)^x. The range of a function encompasses all the possible y-values that the function can produce. To determine the range of our function, we need to consider its behavior as x varies across the entire real number line. We have already established that the function is decreasing, meaning that as x increases, y decreases. As x approaches positive infinity, y gets closer and closer to zero, but as we discussed earlier, it never actually reaches zero. This means that 0 is a lower bound for the range, but it is not included in the range itself. On the other hand, as x approaches negative infinity, the value of (1/3)^x becomes increasingly large, tending towards infinity. This indicates that there is no upper bound for the range. Combining these observations, we can conclude that the range of the function y = (1/3)^x consists of all positive real numbers. In interval notation, this is expressed as (0, ∞). Therefore, to fully evaluate the fifth statement, we would need to know the specific claim it makes about the range. Without the complete statement, we cannot definitively assess its truthfulness. However, based on our analysis, we know that the range is (0, ∞).
To further solidify our understanding of the function y = (1/3)^x, let's visualize its graph. The graph of this function is a smooth curve that starts high on the left side of the coordinate plane and gradually descends towards the x-axis as x increases. The curve approaches the x-axis asymptotically, meaning it gets closer and closer to the x-axis but never actually touches it. This reflects the fact that the function's range is (0, ∞). The graph intersects the y-axis at the point (0, 1), as we determined earlier. The absence of an x-intercept is also evident in the graph, as the curve never crosses the x-axis. The decreasing nature of the function is visually apparent in the downward slope of the curve from left to right. The graph provides a powerful visual representation of the function's properties, reinforcing our analytical findings.
In this comprehensive exploration, we have meticulously examined the function y = (1/3)^x, unraveling its key properties and scrutinizing several statements about its graph. We established that the function is decreasing, has a y-intercept at (0, 1), and its range is (0, ∞). We also debunked the claims that the function is increasing and has an x-intercept at (1, 0). By combining analytical reasoning with graphical visualization, we have gained a deep understanding of the behavior of this exponential decay function. The principles and techniques we have employed in this analysis can be readily applied to other exponential functions, empowering you to confidently explore the fascinating world of mathematical functions and their applications.
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