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

JU07/12/2025 08, 2025 by THE IDEN 40 views

Range of the Function f(x) = -2(6^x) + 3

Understanding the range of a function is crucial in mathematics as it defines the set of all possible output values the function can produce. For the function f(x) = -2(6^x) + 3, we aim to determine this range by analyzing its components and behavior. This involves looking at the exponential part, 6^x, and how it is transformed by the subsequent operations of multiplication by -2 and addition of 3. By carefully considering these transformations, we can accurately identify the range of the function.

Understanding the Exponential Component

At the heart of our function is the exponential term, 6^x. Exponential functions, in general, have the form a^x, where a is a positive constant. In our case, a is 6. Understanding the behavior of exponential functions is fundamental to finding the range of f(x). Let's break down the key characteristics of the exponential function 6^x:

Base Greater Than 1: Since the base 6 is greater than 1, the function 6^x is an increasing function. This means that as x increases, the value of 6^x also increases.
Asymptotic Behavior: As x approaches negative infinity, 6^x approaches 0. However, 6^x never actually reaches 0 because any number (greater than 0) raised to any power will always be greater than 0. Mathematically, we express this as:
```
lim (x→-∞) 6^x = 0
```
Positive Output: For any real number x, the value of 6^x will always be positive. This is a fundamental property of exponential functions with a positive base.
Behavior as x Increases: As x approaches positive infinity, 6^x also approaches positive infinity. This can be written as:
```
lim (x→∞) 6^x = ∞
```

Thus, the range of 6^x is the interval (0, ∞). This means that the output of 6^x can be any positive real number. This understanding forms the foundation for determining the range of f(x).

Transformation by -2

Now that we understand the behavior and range of 6^x, let's consider the transformation applied to it in our function f(x) = -2(6^x) + 3. The first transformation is multiplication by -2. This affects the function in two significant ways:

Vertical Stretch: Multiplying by 2 stretches the function vertically by a factor of 2. This means that every output value of 6^x is doubled.
Reflection over the x-axis: Multiplying by a negative number (-2 in this case) reflects the function over the x-axis. This is a crucial transformation as it changes the direction of the function's range.

Let's analyze the effect on the range. The range of 6^x is (0, ∞). When we multiply by -2, we're essentially multiplying every value in this interval by -2. The key implications are:

Positive Becomes Negative: All positive values in the interval (0, ∞) become negative when multiplied by -2.
Infinity Becomes Negative Infinity: As 6^x approaches infinity, -2(6^x) approaches negative infinity.
Approaches 0 from the Negative Side: As 6^x approaches 0, -2(6^x) also approaches 0, but from the negative side.

Therefore, the range of -2(6^x) is (-∞, 0). This transformation has flipped the range of the exponential function, setting the stage for the final transformation.

Vertical Shift by +3

The final transformation in our function f(x) = -2(6^x) + 3 is the addition of 3. This operation results in a vertical shift of the function. Specifically, adding 3 shifts the entire graph of -2(6^x) upward by 3 units. To determine the impact on the range, we need to consider how this shift affects the interval (-∞, 0), which is the range of -2(6^x).

Shifting the Interval: Adding 3 to every value in the interval (-∞, 0) effectively moves the entire interval 3 units upwards on the number line.
Negative Infinity Remains Negative Infinity: Adding a finite number to negative infinity does not change the nature of negative infinity. Thus, the lower bound of the range remains negative infinity.
0 Shifts to 3: The upper bound of the range, which was 0, shifts to 3 when we add 3.

Therefore, the range of -2(6^x) + 3 is (-∞, 3). This means that the function f(x) can take any value less than 3, but it will never reach or exceed 3. The vertical shift has effectively capped the upper bound of the function's output values.

Determining the Range

To definitively determine the range of the function f(x) = -2(6^x) + 3, we piece together the transformations we've analyzed:

Exponential Function: The base function 6^x has a range of (0, ∞).
Multiplication by -2: Multiplying by -2 reflects the function over the x-axis and stretches it vertically, changing the range to (-∞, 0).
Vertical Shift by +3: Adding 3 shifts the function upward, changing the range to (-∞, 3).

Thus, the range of the function f(x) = -2(6^x) + 3 is the interval (-∞, 3). This means that the output values of the function can be any real number less than 3, but they will never be equal to or greater than 3.

Conclusion

In conclusion, by systematically analyzing the transformations applied to the exponential function 6^x in f(x) = -2(6^x) + 3, we have successfully determined that the range of the function is (-∞, 3). This process highlights the importance of understanding how different operations affect the range of a function. This comprehensive approach not only answers the question but also provides a deep understanding of the underlying principles of function transformations and range determination in mathematics.