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Sagot :
To determine which of the given statements is true about the functions [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], and [tex]\( h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \)[/tex], let's analyze their ranges and domains.
### 1. The range of [tex]\( h(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
Let's start by examining [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
Since [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] produces positive values for all real [tex]\( x \)[/tex] (as it represents an exponential function), multiplying it by [tex]\(-\frac{6}{11}\)[/tex] results in negative values:
[tex]\[ h(x) < 0 \][/tex]
Hence, the range of [tex]\( h(x) \)[/tex] is [tex]\( y < 0 \)[/tex], not [tex]\( y > 0 \)[/tex]. Therefore, this statement is false.
### 2. The domain of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
Next, let's look at [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
The domain of [tex]\( g(x) \)[/tex] involves all real numbers [tex]\( x \)[/tex] because it's defined for any real [tex]\( x \)[/tex]. In this context, [tex]\( y \)[/tex] isn't the domain but rather the variable describing the range of function values. [tex]\( g(x) \)[/tex] produces positive values since [tex]\(\frac{6}{11}\)[/tex] and [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] are positive:
[tex]\[ g(x) > 0 \][/tex]
However, this does not describe the domain of [tex]\( g(x) \)[/tex]. Therefore, this statement is also false.
### 3. The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex].
Let's analyze the ranges:
- For [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^x \][/tex]
Since [tex]\(\left(\frac{11}{2}\right)^x\)[/tex] produces positive values for all real [tex]\( x \)[/tex]:
[tex]\[ f(x) < 0 \][/tex]
- For [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
As previously discussed, [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] produces positive values, hence:
[tex]\[ h(x) < 0 \][/tex]
- For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
Again, [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] produces positive values:
[tex]\[ g(x) > 0 \][/tex]
Therefore, the ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are negative, while the range of [tex]\( g(x) \)[/tex] is positive. Thus, the ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are indeed different from the range of [tex]\( g(x) \)[/tex].
This statement is true.
### 4. The domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are different from the domain of [tex]\( h(x) \)[/tex].
The domain of all the functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] are all real numbers, as they are all defined for any real [tex]\( x \)[/tex]. Thus, the domains are the same across all three functions.
Therefore, this statement is false.
### Conclusion
The correct statement is:
- The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex].
### 1. The range of [tex]\( h(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
Let's start by examining [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
Since [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] produces positive values for all real [tex]\( x \)[/tex] (as it represents an exponential function), multiplying it by [tex]\(-\frac{6}{11}\)[/tex] results in negative values:
[tex]\[ h(x) < 0 \][/tex]
Hence, the range of [tex]\( h(x) \)[/tex] is [tex]\( y < 0 \)[/tex], not [tex]\( y > 0 \)[/tex]. Therefore, this statement is false.
### 2. The domain of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
Next, let's look at [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
The domain of [tex]\( g(x) \)[/tex] involves all real numbers [tex]\( x \)[/tex] because it's defined for any real [tex]\( x \)[/tex]. In this context, [tex]\( y \)[/tex] isn't the domain but rather the variable describing the range of function values. [tex]\( g(x) \)[/tex] produces positive values since [tex]\(\frac{6}{11}\)[/tex] and [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] are positive:
[tex]\[ g(x) > 0 \][/tex]
However, this does not describe the domain of [tex]\( g(x) \)[/tex]. Therefore, this statement is also false.
### 3. The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex].
Let's analyze the ranges:
- For [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^x \][/tex]
Since [tex]\(\left(\frac{11}{2}\right)^x\)[/tex] produces positive values for all real [tex]\( x \)[/tex]:
[tex]\[ f(x) < 0 \][/tex]
- For [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = -\frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
As previously discussed, [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] produces positive values, hence:
[tex]\[ h(x) < 0 \][/tex]
- For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{6}{11}\left(\frac{11}{2}\right)^{-x} \][/tex]
Again, [tex]\(\left(\frac{11}{2}\right)^{-x}\)[/tex] produces positive values:
[tex]\[ g(x) > 0 \][/tex]
Therefore, the ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are negative, while the range of [tex]\( g(x) \)[/tex] is positive. Thus, the ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are indeed different from the range of [tex]\( g(x) \)[/tex].
This statement is true.
### 4. The domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are different from the domain of [tex]\( h(x) \)[/tex].
The domain of all the functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] are all real numbers, as they are all defined for any real [tex]\( x \)[/tex]. Thus, the domains are the same across all three functions.
Therefore, this statement is false.
### Conclusion
The correct statement is:
- The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex].
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