Finding The Range Of F(x) = -2(6^x) + 3

Hey guys! Today, we're diving into a super common question in mathematics: finding the range of a function. Specifically, we're tackling the function f(x) = -2(6^x) + 3. Understanding how to determine the range is crucial for mastering functions, and I'm here to guide you through it step by step. Let's break it down!

Understanding the Function

First, let's understand what the function f(x) = -2(6^x) + 3 represents. This is an exponential function. The key here is recognizing the basic form and how the transformations affect it. The term 6^x is a standard exponential function with a base of 6. This base is greater than 1, which means that as x increases, 6^x also increases exponentially. As x approaches negative infinity, 6^x approaches 0, and as x approaches positive infinity, 6^x approaches infinity. The coefficient -2 in front of 6^x reflects the graph over the x-axis and stretches it vertically by a factor of 2. Finally, adding 3 shifts the entire graph upward by 3 units.

To fully grasp this, consider these components separately. The exponential part, 6^x, always yields a positive value for any real number x. It can get infinitely close to zero but never actually reach it. Therefore, the smallest possible value of 6^x is infinitesimally greater than 0. Now, let's consider the transformation applied to this exponential term. Multiplying by -2 inverts the function and scales it. So, -2(6^x) will always be negative. As x tends to negative infinity, -2(6^x) tends to 0, approaching it from the negative side. As x goes to positive infinity, -2(6^x) tends to negative infinity.

Lastly, we have the vertical shift, +3. This simply moves the entire function up by 3 units. So, whatever values -2(6^x) takes, we add 3 to each of them. As -2(6^x) approaches 0 from the negative side, adding 3 makes the function approach 3 from below. And as -2(6^x) tends to negative infinity, adding 3 doesn't change the fact that the function will still tend to negative infinity. Understanding these transformations is vital for determining the range of the function. When x is a very large negative number, 6^x is close to 0, so -2(6^x) is also close to 0. Adding 3, f(x) is close to 3. When x is a very large positive number, 6^x is a very large positive number, so -2(6^x) is a very large negative number. Adding 3 doesn't change that it's still a very large negative number. Hence, f(x) can take any value smaller than 3.

Determining the Range

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function f(x) = -2(6^x) + 3, we need to determine the possible values of f(x) as x varies across all real numbers. Since 6^x is always positive, -2(6^x) will always be negative. As x approaches negative infinity, 6^x approaches 0, so -2(6^x) approaches 0. Therefore, f(x) approaches 3. However, f(x) will never actually equal 3, because 6^x will never be exactly 0. As x approaches positive infinity, 6^x approaches infinity, so -2(6^x) approaches negative infinity. Therefore, f(x) also approaches negative infinity.

To visualize this, imagine the graph of f(x) = 6^x. It starts very close to the x-axis on the left (as x goes to negative infinity) and rises sharply to the right (as x goes to positive infinity). Now, consider f(x) = -2(6^x). The negative sign flips the graph upside down, so it now starts very close to the x-axis on the left and falls sharply downwards to the right. The 2 stretches the graph vertically. Finally, f(x) = -2(6^x) + 3 shifts the entire graph up by 3 units. This means the graph now approaches y = 3 on the left and falls downwards to negative infinity on the right. Thus, the function can take any value less than 3, but it can never reach 3 or go above it.

Therefore, the range of the function is all real numbers less than 3. In interval notation, this is expressed as (-∞, 3). The function never actually reaches 3 because 6^x is always greater than 0. As x gets more and more negative, 6^x gets closer and closer to zero, making -2(6^x) get closer and closer to zero as well. Adding 3, we get a value that gets closer and closer to 3, but never actually reaches it. On the other end, as x gets more and more positive, 6^x gets larger and larger, so -2(6^x) gets more and more negative. Adding 3 does not change the fact that the value is a large negative number. Therefore, the range of this function includes all numbers less than 3.

Analyzing the Options

Now, let's look at the given options and see which one matches our findings:

A. (-∞, -2] B. (-∞, 3) C. [-2, ∞) D. [3, ∞)

Based on our analysis, the correct range is all real numbers less than 3, which is represented by the interval (-∞, 3). Therefore, option B is the correct answer.

Option A, (-∞, -2], includes all numbers less than or equal to -2. This is incorrect because the function can take values greater than -2, approaching 3.

Option C, [-2, ∞), includes all numbers greater than or equal to -2. This is incorrect because the function can only take values less than 3.

Option D, [3, ∞), includes all numbers greater than or equal to 3. This is incorrect because the function can only take values less than 3.

Conclusion

So, the correct answer is B. (-∞, 3). I hope this explanation clarifies how to find the range of the function f(x) = -2(6^x) + 3. Remember, understanding the transformations applied to the basic exponential function is key to solving these types of problems. Keep practicing, and you'll become a pro at finding ranges in no time!

By understanding exponential functions and their transformations, we were able to determine that the range of f(x) = -2(6^x) + 3 is (-∞, 3). This means that the function can take any value less than 3 but never actually reaches 3.