Understanding the range of functions is crucial in mathematics, particularly when dealing with exponential functions. The range defines the set of all possible output values (y-values) that a function can produce. In this article, we will delve into the analysis of exponential functions to determine which one has a range of y < 3. We will examine each option step-by-step, highlighting key concepts and techniques for identifying the range of exponential functions. This exploration will not only help in answering the specific question but also enhance your overall understanding of function behavior and properties.
Understanding Exponential Functions and Range
Before we dive into the specific options, it's important to understand the fundamental characteristics of exponential functions. An exponential function generally takes the form y = a(b)^x + c, where a, b, and c are constants, and b is the base. The base b is a positive number not equal to 1, and x is the exponent. The constant a determines the vertical stretch or compression and reflection across the x-axis, while c represents the vertical shift.
The range of an exponential function is heavily influenced by the vertical shift (c) and the sign of the coefficient a. When a is positive, the function opens upwards, and when a is negative, the function opens downwards. The horizontal asymptote of the function is given by y = c, which serves as a boundary for the range. For instance, if a is positive, the range will be y > c, and if a is negative, the range will be y < c. Understanding these transformations and their impact on the range is crucial for solving problems involving exponential functions. We will apply these concepts to analyze each of the given options and determine which one satisfies the condition y < 3.
Analyzing Option A: y = 3(2)^x
Let's begin by analyzing the exponential function y = 3(2)^x. In this function, we can identify the parameters as follows: a = 3, b = 2, and c = 0. Since a is positive (3 > 0), the function opens upwards. The base b is 2, which is greater than 1, indicating exponential growth. The vertical shift c is 0, which means the horizontal asymptote is y = 0. As x approaches negative infinity, the term 3(2)^x approaches 0, but it never actually reaches 0. As x approaches positive infinity, the term 3(2)^x grows without bound.
Therefore, the range of the function y = 3(2)^x is y > 0. This is because the function's output will always be a positive number, and it can take any positive value greater than 0. Comparing this to our target range of y < 3, we can see that option A does not satisfy the condition. The values of y for this function are greater than 0, and there is no upper bound, so it cannot be contained within y < 3. Understanding the growth behavior and the impact of the vertical shift is key to correctly determining the range of this function. This analysis allows us to confidently eliminate option A as the correct answer.
Analyzing Option B: y = 2(3)^x
Next, let's consider the exponential function y = 2(3)^x. Similar to the previous function, we identify the parameters as a = 2, b = 3, and c = 0. Here, a is positive (2 > 0), so the function opens upwards. The base b is 3, which is greater than 1, signifying exponential growth. The vertical shift c is 0, meaning the horizontal asymptote is y = 0. As x decreases towards negative infinity, the value of 2(3)^x approaches 0 but never actually equals 0. As x increases towards positive infinity, the value of 2(3)^x grows without limit.
Consequently, the range of the function y = 2(3)^x is y > 0. This function's output values are always positive and can take any value greater than 0. This means that the range does not satisfy the condition y < 3. The function values are bounded below by 0, but there is no upper bound, and they are all positive. Therefore, option B can also be eliminated. Recognizing the exponential growth and the effect of the vertical asymptote is essential in accurately identifying the range of the function. This detailed analysis helps us to narrow down the options and focus on the ones that might satisfy the condition y < 3.
Analyzing Option C: y = -(2)^x + 3
Now, let's examine the exponential function y = -(2)^x + 3. In this case, the parameters are a = -1, b = 2, and c = 3. The negative sign in front of the exponential term indicates that a is negative (-1 < 0), so the function opens downwards. The base b is 2, which is greater than 1, indicating exponential decay from the top down. The vertical shift c is 3, which means the horizontal asymptote is y = 3. As x approaches negative infinity, the term -(2)^x approaches 0, so y approaches 3 from below. As x approaches positive infinity, the term -(2)^x becomes increasingly negative, causing y to decrease without bound.
Thus, the range of the function y = -(2)^x + 3 is y < 3. This is because the function's output values are always less than 3. The negative coefficient a reflects the function across the x-axis, causing it to open downwards, and the vertical shift c = 3 sets the upper bound for the range. This function precisely matches the condition we are looking for. Therefore, option C is a potential answer. Understanding the impact of the negative coefficient and the vertical shift is crucial for determining the range of this function. This detailed analysis confirms that option C satisfies the given condition.
Analyzing Option D: y = (2)^x - 3
Finally, let's analyze the exponential function y = (2)^x - 3. The parameters for this function are a = 1, b = 2, and c = -3. Here, a is positive (1 > 0), so the function opens upwards. The base b is 2, which is greater than 1, indicating exponential growth. The vertical shift c is -3, which means the horizontal asymptote is y = -3. As x approaches negative infinity, the term (2)^x approaches 0, so y approaches -3 but never actually reaches it. As x approaches positive infinity, the term (2)^x grows without bound, causing y to increase without limit.
Therefore, the range of the function y = (2)^x - 3 is y > -3. This means the function's output values are always greater than -3. Comparing this to our target range of y < 3, we can see that option D does not satisfy the condition. The values of y for this function are greater than -3, and there is no upper bound within the range y < 3. Understanding the growth behavior and the impact of the vertical shift is key to correctly determining the range of this function. This analysis allows us to confidently eliminate option D as the correct answer.
Conclusion: Identifying the Function with Range y < 3
After a thorough analysis of all the options, we can conclude that the function with a range of y < 3 is option C: y = -(2)^x + 3. We examined the characteristics of each exponential function, paying close attention to the coefficients, bases, and vertical shifts. Option C was the only function where the negative coefficient in front of the exponential term and the vertical shift resulted in a range bounded above by 3. This comprehensive analysis highlights the importance of understanding the properties of exponential functions and how different parameters affect their behavior.
The process involved understanding the basic form of an exponential function, y = a(b)^x + c, and recognizing how the constants a, b, and c influence the function's graph and range. By determining whether the function opens upwards or downwards and identifying the horizontal asymptote, we could accurately determine the range for each option. This methodical approach is crucial for solving similar problems and for building a strong foundation in mathematical analysis. Ultimately, this exercise not only answers the specific question but also reinforces the broader concepts of function analysis and transformations.