Analyzing Ranges Of Exponential Functions F(x), G(x), And H(x)

$f(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^x \quad g(x)=\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \quad h(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^{-x}$

To delve into the characteristics of these exponential functions, we must meticulously examine their ranges, which dictate the possible output values. Exponential functions, in their general form, exhibit unique behaviors contingent on their base and coefficient. The base, in this case, ${\frac{11}{2}}$, is greater than 1, indicating exponential growth when the exponent is positive and exponential decay when the exponent is negative. The coefficient, whether positive or negative, plays a pivotal role in determining the reflection of the function across the x-axis and, consequently, its range.

Analyzing f(x), we observe a negative coefficient (- ${\frac{6}{11}}$) and a base greater than 1 raised to the power of x. As x increases, ${\left(\frac{11}{2}\right)^x}$ grows exponentially. However, the negative coefficient reflects the function across the x-axis, causing it to approach negative infinity as x increases and approach 0 from the negative side as x decreases. Consequently, the range of f(x) is all negative real numbers, expressed as (-∞, 0).

Turning our attention to g(x), we encounter a positive coefficient (${\frac{6}{11}}$) and a base greater than 1 raised to the power of -x. The negative exponent signifies exponential decay, meaning that as x increases, ${\left(\frac{11}{2}\right)^{-x}}$ approaches 0. Conversely, as x decreases, ${\left(\frac{11}{2}\right)^{-x}}$ grows exponentially. Since the coefficient is positive, the function remains above the x-axis, approaching 0 as x increases and positive infinity as x decreases. Therefore, the range of g(x) is all positive real numbers, represented as (0, ∞).

Lastly, let's dissect h(x). It presents a negative coefficient (- ${\frac{6}{11}}$) and a base greater than 1 raised to the power of -x. Similar to g(x), the negative exponent denotes exponential decay. However, the negative coefficient introduces a reflection across the x-axis. As x increases, ${\left(\frac{11}{2}\right)^{-x}}$ approaches 0, and due to the negative coefficient, h(x) also approaches 0 from the negative side. As x decreases, ${\left(\frac{11}{2}\right)^{-x}}$ grows exponentially, but the negative coefficient causes h(x) to approach negative infinity. Thus, the range of h(x) is all negative real numbers, expressed as (-∞, 0).

In summary, the ranges of the functions are as follows:

• f(x): (-∞, 0)
• g(x): (0, ∞)
• h(x): (-∞, 0)

Understanding the interplay between the base, exponent, and coefficient is crucial for accurately determining the range of exponential functions. The sign of the coefficient dictates the reflection across the x-axis, while the base and exponent govern the growth or decay behavior of the function.

Determining the Correct Statement

To identify the true statement, we need to analyze the ranges of the given functions: $f(x)$, $g(x)$, and $h(x)$. As previously established, exponential functions exhibit distinct behaviors based on their base, exponent, and coefficient. Let's dissect each function individually to ascertain its range and subsequently evaluate the provided statements.

In-Depth Analysis of f(x)

Focusing on $f(x) = -\frac{6}{11}(\frac{11}{2})^x$, we observe a negative coefficient ($-\frac{6}{11}$) and a base greater than 1 ($\frac{11}{2}$) raised to the power of x. The base, being greater than 1, signifies exponential growth. However, the negative coefficient introduces a critical transformation: a reflection across the x-axis. As x increases, $(\frac{11}{2})^x$ grows exponentially, but the negative coefficient flips this growth, causing the function to approach negative infinity. Conversely, as x decreases, $(\frac{11}{2})^x$ approaches 0, and due to the negative coefficient, $f(x)$ approaches 0 from the negative side. This behavior confines the range of $f(x)$ to all negative real numbers, excluding 0. Mathematically, this range is expressed as (-∞, 0).

Detailed Examination of g(x)

Now, let's turn our attention to $g(x) = \frac{6}{11}(\frac{11}{2})^{-x}$. This function features a positive coefficient ($\frac{6}{11}$) and a base greater than 1 ($\frac{11}{2}$) raised to the power of -x. The negative exponent is the key here, indicating exponential decay. As x increases, $(\frac{11}{2})^{-x}$ approaches 0. As x decreases, $(\frac{11}{2})^{-x}$ grows exponentially. The positive coefficient ensures that the function remains above the x-axis. Therefore, $g(x)$ approaches 0 as x increases and approaches positive infinity as x decreases. This defines the range of $g(x)$ as all positive real numbers, represented as (0, ∞).

Comprehensive Understanding of h(x)

Finally, let's scrutinize $h(x) = -\frac{6}{11}(\frac{11}{2})^{-x}$. This function combines a negative coefficient ($-\frac{6}{11}$) with a base greater than 1 ($\frac{11}{2}$) raised to the power of -x. The negative exponent, as in $g(x)$, signifies exponential decay. However, the negative coefficient introduces a reflection across the x-axis, similar to $f(x)$. As x increases, $(\frac{11}{2})^{-x}$ approaches 0, and due to the negative coefficient, $h(x)$ approaches 0 from the negative side. As x decreases, $(\frac{11}{2})^{-x}$ grows exponentially, but the negative coefficient causes $h(x)$ to approach negative infinity. Consequently, the range of $h(x)$ encompasses all negative real numbers, denoted as (-∞, 0).

Synthesizing the Range Analysis

To recap, we've meticulously analyzed the ranges of the three functions:

• $f(x)$: (-∞, 0)
• $g(x)$: (0, ∞)
• $h(x)$: (-∞, 0)

This comprehensive understanding forms the bedrock for accurately evaluating the statements and pinpointing the true one. By recognizing the influence of coefficients and exponents on the behavior of exponential functions, we can confidently navigate the complexities of their ranges.

Identifying the Correct Answer

After thoroughly analyzing the ranges of the functions $f(x)$, $g(x)$, and $h(x)$, we are now equipped to determine the correct statement. Our detailed examination revealed the following ranges:

• $f(x) = -\frac{6}{11}(\frac{11}{2})^x$: Range is (-∞, 0)
• $g(x) = \frac{6}{11}(\frac{11}{2})^{-x}$: Range is (0, ∞)
• $h(x) = -\frac{6}{11}(\frac{11}{2})^{-x}$: Range is (-∞, 0)

To pinpoint the accurate statement, we need to carefully compare these ranges and identify which statement aligns with our findings. It's crucial to meticulously analyze each option, ensuring it accurately reflects the behavior of the functions and their respective ranges. Any discrepancy between the stated range and our calculated range would immediately disqualify that statement.

Strategic Approach to Statement Evaluation

When tackling such problems, a strategic approach is paramount. Rather than blindly evaluating each statement, it's often beneficial to identify potential areas of focus. For instance, statements that make broad generalizations about all three functions might be more susceptible to error than those focusing on a single function. Similarly, statements that involve comparisons between ranges should be examined with particular scrutiny.

Consider the following strategy:

1. Focus on Key Differences: Highlight the key distinctions between the functions. In this case, the presence or absence of a negative sign in the coefficient and the exponent significantly impacts the function's behavior and range.
2. Eliminate Incorrect Statements: Start by eliminating statements that are demonstrably false. This narrows down the options and increases the likelihood of selecting the correct answer.
3. Verify the Remaining Statements: For the remaining statements, meticulously verify their accuracy against the calculated ranges. Ensure that the statement precisely reflects the range of each function.
4. Pay Attention to Nuances: Be mindful of subtle differences in wording. A single word can alter the meaning of a statement and affect its validity.

Practical Application of the Strategy

Let's illustrate this strategy with a hypothetical example. Suppose one of the statements claims that the range of $f(x)$ is all real numbers. We can immediately eliminate this statement because our analysis clearly demonstrates that the range of $f(x)$ is limited to negative real numbers. This simple step eliminates one option and streamlines the process of finding the correct answer.

Similarly, if another statement asserts that the ranges of $g(x)$ and $h(x)$ are identical, we can quickly dismiss it. Our findings reveal that the range of $g(x)$ is positive real numbers, while the range of $h(x)$ is negative real numbers. These ranges are mutually exclusive, rendering the statement false.

Conclusive Determination of the Truth

By systematically applying this strategic approach, we can efficiently navigate the options and arrive at the true statement. The key lies in leveraging our comprehensive understanding of the functions' ranges and meticulously comparing them against the claims made in each statement. This methodical process ensures that we select the statement that accurately represents the relationships between the functions and their ranges.

Consider the functions $f(x)=-\frac{6}{11}(\frac{11}{2})^x$, $g(x)=\frac{6}{11}(\frac{11}{2})^{-x}$, and $h(x)=-\frac{6}{11}(\frac{11}{2})^{-x}$. Which statement about their ranges is true?

Analyzing Ranges of Exponential Functions f(x), g(x), and h(x)