Analyzing Exponential Function F(x) = 3(1/3)^x And Its Graph

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Let's dive into an analysis of the exponential function f(x) = 3(1/3)^x and explore its graph properties. This function is a classic example of exponential decay, and understanding its behavior is crucial for various applications in mathematics, science, and finance. We'll break down the key characteristics of this function and identify the correct statements about its graph.

Understanding the Exponential Function

When analyzing an exponential function, it's essential to consider its general form, which is typically f(x) = ab^x*, where a is the initial value and b is the base. In our case, f(x) = 3(1/3)^x, we can see that a = 3 and b = 1/3. The base b plays a crucial role in determining whether the function represents exponential growth or decay. If b > 1, the function represents exponential growth, while if 0 < b < 1, it signifies exponential decay. Since our base is 1/3, which falls between 0 and 1, we know that f(x) is an exponential decay function.

The initial value, a = 3, represents the y-intercept of the graph. This means that when x = 0, the function's value is f(0) = 3(1/3)^0 = 3. So, the graph will pass through the point (0, 3). The exponential function's behavior as x increases is also significant. As x becomes larger and larger, the term (1/3)^x gets smaller and smaller, approaching zero. This means that the function's values will decrease, getting closer and closer to the x-axis but never actually touching it. This horizontal line that the function approaches is called a horizontal asymptote, and for this function, the horizontal asymptote is the x-axis, y = 0.

Furthermore, let's consider the transformation aspect of the function. The function f(x) = 3(1/3)^x can be seen as a transformation of the basic exponential decay function g(x) = (1/3)^x. The multiplication by 3 in f(x) represents a vertical stretch by a factor of 3. This means that every y-value of g(x) is multiplied by 3 to obtain the corresponding y-value of f(x). This stretching effect makes the graph of f(x) appear steeper compared to the graph of g(x). In summary, this exponential function exhibits key features of decay, with a starting value of 3 and a gradual decrease towards the x-axis as x increases. Recognizing these aspects is key to accurately interpreting its graph and behavior. This function gives us a clear example of how exponential functions can model real-world phenomena where quantities decrease over time, such as radioactive decay or the depreciation of an asset.

Analyzing the Graph of f(x) = 3(1/3)^x

The graph of the exponential function f(x) = 3(1/3)^x provides a visual representation of its behavior. As we discussed earlier, this function exhibits exponential decay, and the graph clearly reflects this characteristic. The curve starts at the point (0, 3), which is the y-intercept, and gradually decreases as x increases. The graph approaches the x-axis (y = 0) but never actually intersects it, demonstrating the presence of a horizontal asymptote at y = 0. This asymptotic behavior is a hallmark of exponential functions, indicating that the function's values get arbitrarily close to zero as x tends towards infinity.

The absence of any x-intercept is another key feature of the graph. Since the function's values are always positive (due to the positive initial value and the base between 0 and 1), the graph will never cross the x-axis. This also means that the function has no real roots or zeros. The domain of the function is all real numbers, as we can input any value of x into the function. However, the range is limited to positive values, specifically y > 0, because the function's output is always greater than zero. The curve is smooth and continuous, indicating that the function is defined for all real numbers. As we move from left to right along the graph, we observe a decreasing trend, further confirming that this is an exponential decay function. The rate of decay is determined by the base (1/3), which indicates that the function's value decreases by a factor of 1/3 for every unit increase in x.

Compared to the basic exponential decay function g(x) = (1/3)^x, the graph of f(x) is vertically stretched by a factor of 3. This means that the graph of f(x) is further away from the x-axis than the graph of g(x) for any given value of x. This stretching effect is a direct result of the multiplication by 3 in the function's equation. In summary, the graph of f(x) = 3(1/3)^x visually represents exponential decay, showing a decreasing curve that approaches the x-axis, with a y-intercept at (0, 3) and a horizontal asymptote at y = 0. Understanding these graphical features helps in quickly recognizing and interpreting the behavior of this type of exponential function.

Key Properties and Statements About f(x) = 3(1/3)^x

When evaluating statements about the function f(x) = 3(1/3)^x and its graph, it is crucial to consider the key properties we have already discussed. One of the fundamental aspects of this function is that it represents exponential decay. This means that as x increases, the function's value decreases. This characteristic is directly related to the base of the exponential term, which is 1/3 in this case. Since 0 < 1/3 < 1, the function exhibits decay rather than growth.

Another important property to consider is the y-intercept. As we established, the y-intercept occurs when x = 0, and for this function, f(0) = 3(1/3)^0 = 3. Therefore, the graph of the function intersects the y-axis at the point (0, 3). The horizontal asymptote is also a critical feature. For exponential functions of this form, the horizontal asymptote is the x-axis, y = 0. This means that the function's values get closer and closer to zero as x becomes very large, but the graph never actually touches or crosses the x-axis.

Furthermore, it's important to recognize the transformation that has been applied to the basic exponential decay function g(x) = (1/3)^x. The function f(x) is a vertical stretch of g(x) by a factor of 3. This vertical stretch affects the y-values of the function, making the graph of f(x) steeper than that of g(x). Additionally, the domain of the function f(x) is all real numbers, as we can input any value of x into the function. However, the range is limited to positive values, y > 0, since the function's output is always positive due to the initial value of 3 and the base between 0 and 1.

Now, let's consider some specific statements that might be made about this function and its graph. A statement claiming that the function represents exponential growth would be incorrect, as we have established that it is an exponential decay function. A statement indicating that the y-intercept is at a different point, such as (0, 1), would also be incorrect. Similarly, a statement suggesting that the horizontal asymptote is not the x-axis would be false. Correct statements would include that the function is a vertical stretch of g(x) = (1/3)^x by a factor of 3, that the y-intercept is (0, 3), and that the horizontal asymptote is y = 0. Evaluating these properties and understanding the function's behavior are crucial for accurately determining the validity of different statements.

Choosing the Correct Statements

To accurately identify the true statements about the exponential function f(x) = 3(1/3)^x and its graph, we need to carefully consider its properties and characteristics. As we've already discussed, this function is a prime example of exponential decay, and this understanding is key to evaluating the validity of different claims. The fact that the base (1/3) is between 0 and 1 immediately tells us that the function's values will decrease as x increases.

One crucial aspect to consider is the y-intercept. We've established that f(0) = 3(1/3)^0 = 3, which means the graph intersects the y-axis at the point (0, 3). Any statement suggesting a different y-intercept would be incorrect. Similarly, the horizontal asymptote is a defining feature of this exponential function. Since the function's values approach the x-axis (y = 0) as x becomes large, the horizontal asymptote is y = 0. Any statement claiming a different asymptote would be false.

Another critical element to evaluate is the transformation applied to the basic exponential function g(x) = (1/3)^x. The function f(x) = 3(1/3)^x is a vertical stretch of g(x) by a factor of 3. This means that the y-values of f(x) are three times the y-values of g(x) for any given x. Therefore, statements correctly identifying this vertical stretch would be accurate.

Considering these points, let's analyze some potential statements:

  • A statement claiming "The function is a stretch of the function g(x) = (1/3)^x" is likely to be true, as we've determined that f(x) is a vertical stretch of g(x).
  • A statement asserting "The y-intercept is (0, 3)" is also true, as we've calculated and confirmed this.
  • A statement indicating "The graph has a horizontal asymptote at y = 0" is correct, based on our understanding of the function's behavior as x increases.

On the other hand, statements claiming:

  • "The function represents exponential growth" would be false.
  • "The y-intercept is (0, 1)" would be incorrect.
  • "The horizontal asymptote is y = 1" would be inaccurate.

By carefully considering the function's properties, including its decay behavior, y-intercept, horizontal asymptote, and transformations, we can confidently select the true statements about f(x) = 3(1/3)^x and its graph. This systematic approach ensures accuracy and a deep understanding of exponential functions.

In conclusion, analyzing exponential functions like f(x) = 3(1/3)^x involves understanding their key characteristics, such as exponential decay, y-intercept, horizontal asymptote, and transformations. By carefully considering these properties, we can accurately interpret the function's graph and identify true statements about its behavior.