Answer :
To answer the question, we need to analyze the domains and ranges of the given functions:
The functions are:
[tex]\[ f(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^x \][/tex]
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
### Step-by-Step Analysis:
1. Range of [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
- The term [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] is always positive for all real [tex]\(x\)[/tex].
- Multiplying a positive number by [tex]\(-\frac{6}{11}\)[/tex] makes the result negative.
Therefore, the range of [tex]\( h(x) \)[/tex] is [tex]\( y < 0 \)[/tex]. The statement "The range of [tex]\( h(x) \)[/tex] is [tex]\( y > 0 \)[/tex]" is false.
2. Domain of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
- The expression [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] is well-defined for all real [tex]\(x\)[/tex].
Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers ([tex]\( \mathbb{R} \)[/tex]). The statement "The domain of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex]" is false (this statement incorrectly refers to the domain as if it were describing a range).
3. Range of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^x\][/tex]
- The term [tex]\(\left(\frac{11}{2}\right)^x\)[/tex] is always positive for all real [tex]\(x\)[/tex].
- Multiplying a positive number by [tex]\(-\frac{6}{11}\)[/tex] makes the result negative.
Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\( y < 0 \)[/tex].
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
- The term [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] is always positive for all real [tex]\(x\)[/tex].
- Multiplying a positive number by [tex]\(\frac{6}{11}\)[/tex] makes the result positive.
Therefore, the range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
Given that both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have ranges [tex]\( y < 0 \)[/tex], and [tex]\( g(x) \)[/tex] has [tex]\( y > 0 \)[/tex], the ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex]. The statement "The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex]" is true.
4. Domain of [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex]:
- All three functions use [tex]\( x \)[/tex] in an exponent and do not impose any restrictions on [tex]\( x \)[/tex].
Therefore, the domains of [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] are all real numbers ([tex]\( \mathbb{R} \)[/tex]). The statement "The domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are different from the domain of [tex]\( h(x) \)[/tex]" is false.
### Conclusion:
The true statement is:
"The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex]."
The functions are:
[tex]\[ f(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^x \][/tex]
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
### Step-by-Step Analysis:
1. Range of [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
- The term [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] is always positive for all real [tex]\(x\)[/tex].
- Multiplying a positive number by [tex]\(-\frac{6}{11}\)[/tex] makes the result negative.
Therefore, the range of [tex]\( h(x) \)[/tex] is [tex]\( y < 0 \)[/tex]. The statement "The range of [tex]\( h(x) \)[/tex] is [tex]\( y > 0 \)[/tex]" is false.
2. Domain of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
- The expression [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] is well-defined for all real [tex]\(x\)[/tex].
Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers ([tex]\( \mathbb{R} \)[/tex]). The statement "The domain of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex]" is false (this statement incorrectly refers to the domain as if it were describing a range).
3. Range of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^x\][/tex]
- The term [tex]\(\left(\frac{11}{2}\right)^x\)[/tex] is always positive for all real [tex]\(x\)[/tex].
- Multiplying a positive number by [tex]\(-\frac{6}{11}\)[/tex] makes the result negative.
Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\( y < 0 \)[/tex].
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
- The term [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] is always positive for all real [tex]\(x\)[/tex].
- Multiplying a positive number by [tex]\(\frac{6}{11}\)[/tex] makes the result positive.
Therefore, the range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
Given that both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] have ranges [tex]\( y < 0 \)[/tex], and [tex]\( g(x) \)[/tex] has [tex]\( y > 0 \)[/tex], the ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex]. The statement "The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex]" is true.
4. Domain of [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex]:
- All three functions use [tex]\( x \)[/tex] in an exponent and do not impose any restrictions on [tex]\( x \)[/tex].
Therefore, the domains of [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] are all real numbers ([tex]\( \mathbb{R} \)[/tex]). The statement "The domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are different from the domain of [tex]\( h(x) \)[/tex]" is false.
### Conclusion:
The true statement is:
"The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex]."
