Consider the three functions below. $$ f(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^{x} \quad g(x)=\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \quad h(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^{-x} $$ Which statement is true? The range of $$ h(x) $$ is $$ y0 $$. The domain of $$ g(x) $$ is $$ y0 $$. The ranges of $$ f(x) $$ and $$ h(x) $$ are different from the range of $$ g(x) $$. The domains of $$ f(x) $$ and $$ g(x) $$ are different from the domain of $$ h(x) $$.

Question

Consider the three functions below. $$ f(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^{x} \quad g(x)=\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \quad h(x)=-\frac{6}{11}\left(\frac{11}{2}\right)^{-x} $$ Which statement is true? The range of $$ h(x) $$ is $$ y>0 $$. The domain of $$ g(x) $$ is $$ y>0 $$. The ranges of $$ f(x) $$ and $$ h(x) $$ are different from the range of $$ g(x) $$. The domains of $$ f(x) $$ and $$ g(x) $$ are different from the domain of $$ h(x) $$.

UpStudy AI · Accepted Answer

To analyze the functions $$ f(x) $$, $$ g(x) $$, and $$ h(x) $$, we will look at their forms and determine their ranges and domains.

1. **Function $$ f(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{x} $$**:
   - This function is an exponential function multiplied by a negative constant. As $$ x $$ increases, $$ \left(\frac{11}{2}\right)^{x} $$ increases, making $$ f(x) $$ decrease towards negative infinity. As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches 0 from the negative side.
   - **Range of $$ f(x) $$**: $$ (-\infty, 0) $$

2. **Function $$ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} $$**:
   - This function can be rewritten as $$ g(x) = \frac{6}{11}\left(\frac{2}{11}\right)^{x} $$. As $$ x $$ increases, $$ \left(\frac{2}{11}\right)^{x} $$ decreases towards 0, and as $$ x $$ approaches negative infinity, $$ g(x) $$ approaches positive infinity.
   - **Range of $$ g(x) $$**: $$ (0, \infty) $$

3. **Function $$ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} $$**:
   - This function can be rewritten as $$ h(x) = -\frac{6}{11}\left(\frac{2}{11}\right)^{-x} $$. As $$ x $$ increases, $$ \left(\frac{2}{11}\right)^{-x} $$ increases, making $$ h(x) $$ decrease towards negative infinity. As $$ x $$ approaches negative infinity, $$ h(x) $$ approaches 0 from the negative side.
   - **Range of $$ h(x) $$**: $$ (-\infty, 0) $$

Now, let's evaluate the statements:

1. **The range of $$ h(x) $$ is $$ y > 0 $$**: **False**. The range of $$ h(x) $$ is $$ (-\infty, 0) $$.

2. **The domain of $$ g(x) $$ is $$ y > 0 $$**: **False**. The domain of $$ g(x) $$ is all real numbers $$ (-\infty, \infty) $$.

3. **The ranges of $$ f(x) $$ and $$ h(x) $$ are different from the range of $$ g(x) $$**: **True**. The ranges of $$ f(x) $$ and $$ h(x) $$ are $$ (-\infty, 0) $$, while the range of $$ g(x) $$ is $$ (0, \infty) $$.

4. **The domains of $$ f(x) $$ and $$ g(x) $$ are different from the domain of $$ h(x) $$**: **False**. All three functions have the same domain, which is all real numbers $$ (-\infty, \infty) $$.

Thus, the true statement is:

**The ranges of $$ f(x) $$ and $$ h(x) $$ are different from the range of $$ g(x) $$.**
The true statement is: The ranges of $$ f(x) $$ and $$ h(x) $$ are different from the range of $$ g(x) $$.

Consider the three functions below.

Which statement is true?
The range of is .
The domain of is .
The ranges of and are different from the range of .
The domains of and are different from the domain of .

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Consider the three functions below. Which statement is true? The range of is . The domain of is . The ranges of and are different from the range of . The domains of and are different from the domain of .