Identifying Functions With A Range Of Y Less Than 3

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Hey guys! Today, we're diving into the fascinating world of functions and their ranges, specifically focusing on which function among a given set has a range of y < 3. This means we're looking for a function where the output values (y) are always less than 3. Understanding the range of a function is super important in mathematics as it tells us the set of all possible output values we can get from that function. It helps us visualize the function's behavior and predict its outcomes. So, let's put on our math hats and explore these functions together!

Understanding Function Ranges

Before we jump into the specific functions, let's quickly recap what the range of a function actually means. The range is essentially the set of all possible y-values that a function can produce. Think of it as the vertical spread of the function's graph. When we say y < 3, we're specifying that the function's output values must be less than 3, but not equal to 3. This is often seen in exponential functions with transformations, where the horizontal asymptote plays a key role in defining the range. Grasping this concept is crucial, as it sets the foundation for our analysis. We need to visualize how different functions behave and what limits their output values. Now that we've refreshed our understanding of ranges, we're better equipped to tackle the given functions and pinpoint the one that fits our y < 3 criteria. Remember, the key is to think about how the function's equation translates to its graph and, consequently, its range. Let's get to it!

Analyzing the Functions

We are given four functions, and our mission is to determine which one has a range of y < 3. Let's break them down one by one:

1. y = 3(2)^x

  • Initial Analysis: This is an exponential function. Exponential functions generally have a horizontal asymptote at y = 0. However, the multiplication by 3 stretches the function vertically.
  • Detailed Explanation: The base of the exponent is 2, which is greater than 1, indicating exponential growth. As x increases, 2^x increases, and when multiplied by 3, it grows even faster. The function 3(2)^x will always produce positive values. As x approaches negative infinity, 2^x approaches 0, so 3(2)^x approaches 0. This means the horizontal asymptote is at y = 0. Since the function is always positive and grows exponentially, its range is y > 0. The range can be written in interval notation as (0, ∞). Because the function only takes on values greater than 0, it will never have a value less than 3. Therefore, y = 3(2)^x does not satisfy the condition y < 3.

2. y = 2(3)^x

  • Initial Analysis: This is another exponential function, similar to the first one, but with a different base and coefficient.
  • Detailed Explanation: Again, we have exponential growth since the base 3 is greater than 1. The coefficient 2 stretches the graph vertically. The function 2(3)^x will always produce positive values. As x approaches negative infinity, 3^x approaches 0, so 2(3)^x approaches 0. This means the horizontal asymptote is at y = 0. The range is y > 0, which in interval notation is (0, ∞). Similar to the first function, this function's output values are always greater than 0, so it never has a value less than 3. Thus, y = 2(3)^x does not fit the y < 3 requirement.

3. y = -(2)^x + 3

  • Initial Analysis: This function includes a negative sign in front of the exponential term and a vertical shift. This is a crucial difference that could affect the range.
  • Detailed Explanation: The negative sign in front of 2^x reflects the graph across the x-axis. Without the +3, the function -2^x would approach 0 from below as x approaches negative infinity, and would decrease without bound as x increases. The +3 shifts the entire graph upwards by 3 units. This means the horizontal asymptote shifts from y = 0 to y = 3. Because of the reflection, the function will always be below the asymptote. As x approaches negative infinity, -2^x approaches 0, so -(2)^x + 3 approaches 3. The function values are always less than 3, so the range is y < 3. In interval notation, this is (-∞, 3). This function does satisfy our condition of y < 3. The reflection and vertical shift combine to create a range that is strictly less than 3, making it our potential answer.

4. y = (2)^x - 3

  • Initial Analysis: This is an exponential function with a vertical shift downwards.
  • Detailed Explanation: The base 2 indicates exponential growth. The -3 shifts the entire graph downwards by 3 units. The horizontal asymptote shifts from y = 0 to y = -3. As x approaches negative infinity, 2^x approaches 0, so (2)^x - 3 approaches -3. The function values are always greater than -3, so the range is y > -3. In interval notation, this is (-3, ∞). While this function's values can be less than 3, they are not always less than 3. Therefore, y = (2)^x - 3 does not have a range of y < 3.

The Verdict

After carefully analyzing each function, we've found that only one of them has a range of y < 3. Guys, it's...

The function y = -(2)^x + 3 has a range of y < 3.

The negative sign reflects the exponential function across the x-axis, and the +3 shifts the graph upwards by 3 units, resulting in a range where all y-values are less than 3. This function perfectly fits the criteria we were looking for!

Key Takeaways

This exercise has highlighted some key concepts in understanding function ranges. Remember these important points:

  • Exponential Functions: Exponential functions of the form y = a(b)^x have a horizontal asymptote at y = 0 if there are no vertical shifts. The value of b determines whether the function grows (b > 1) or decays (0 < b < 1).
  • Vertical Shifts: Adding or subtracting a constant from a function shifts the graph vertically. A positive constant shifts the graph upwards, and a negative constant shifts it downwards.
  • Reflections: A negative sign in front of the function reflects the graph across the x-axis. This can drastically change the range of the function.
  • Range and Asymptotes: The horizontal asymptote is a crucial element in determining the range of exponential functions. The range will either be all values greater than the asymptote or all values less than the asymptote, depending on the presence of reflections.

By keeping these concepts in mind, you'll be well-equipped to analyze the ranges of various functions and solve similar problems. Great job, everyone! Keep up the awesome work!

Practice Problems

To solidify your understanding, try these practice problems:

  1. Which function has a range of y > 5?

    • y = 2(3)^x + 5
    • y = -(2)^x + 5
    • y = (1/2)^x + 5
    • y = -(1/2)^x + 5

  2. Determine the range of the function y = -3(2)^x - 1.

  3. Which of the following functions has a range that includes negative values?

    • y = 5^x
    • y = 5^x + 2
    • y = -5^x
    • y = -5^x + 10

Working through these problems will help you build confidence and expertise in analyzing function ranges. Happy solving!