To solve the given problem, let's start by analyzing the functions one by one.
1. Function [tex]\( f(x) = -\frac{6}{11} \left( \frac{11}{2} \right)^x \)[/tex]
- Domain: Exponential functions are defined for all real numbers [tex]\( x \)[/tex]. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers, i.e., [tex]\( x \in \mathbb{R} \)[/tex].
- Range: The base [tex]\( \left( \frac{11}{2} \right) \)[/tex] is a positive number greater than 1. As [tex]\( x \)[/tex] varies over all real numbers, [tex]\( \left( \frac{11}{2} \right)^x \)[/tex] takes all positive values. However, since we are multiplying by a negative constant [tex]\( -\frac{6}{11} \)[/tex], the values of [tex]\( f(x) \)[/tex] will be all negative real numbers. Hence, the range of [tex]\( f(x) \)[/tex] is [tex]\( y < 0 \)[/tex].
2. Function [tex]\( g(x) = \frac{6}{11} \left( \frac{11}{2} \right)^{-x} \)[/tex]
- Domain: Again, exponential functions are defined for all real numbers [tex]\( x \)[/tex]. Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers, i.e., [tex]\( x \in \mathbb{R} \)[/tex].
- Range: When the exponent is negated, [tex]\( \left( \frac{11}{2} \right)^{-x} \)[/tex] becomes a fraction that is always positive. Multiplying by the positive constant [tex]\( \frac{6}{11} \)[/tex] keeps the function [tex]\( g(x) \)[/tex] positive. Hence, the range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
3. Function [tex]\( h(x) = -\frac{6}{11} \left( \frac{11}{2} \right)^{-x} \)[/tex]
- Domain: Exponential functions are defined for all real numbers [tex]\( x \)[/tex]. Therefore, the domain of [tex]\( h(x) \)[/tex] is all real numbers, i.e., [tex]\( x \in \mathbb{R} \)[/tex].
- Range: Similar to [tex]\( g(x) \)[/tex], the term [tex]\( \left( \frac{11}{2} \right)^{-x} \)[/tex] is positive. However, because we are multiplying by a negative constant [tex]\( -\frac{6}{11} \)[/tex], the values of [tex]\( h(x) \)[/tex] will be all negative real numbers. Hence, the range of [tex]\( h(x) \)[/tex] is [tex]\( y < 0 \)[/tex].
Now, let's evaluate the given statements:
1. The range of [tex]\( h(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
- This statement is false. We determined that the range of [tex]\( h(x) \)[/tex] is [tex]\( y < 0 \)[/tex].
2. The domain of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
- This statement is incorrect. The domain refers to the possible values of [tex]\( x \)[/tex], not [tex]\( y \)[/tex]. The domain of [tex]\( g(x) \)[/tex] is all real numbers, not [tex]\( y > 0 \)[/tex].
3. The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are different from the range of [tex]\( g(x) \)[/tex].
- This statement is true. The ranges of [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex] are [tex]\( y < 0 \)[/tex], while the range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
4. The domains of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are different from the domain of [tex]\( h(x) \)[/tex].
- This statement is false. The domains of all three functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex] are the same: all real numbers.
Hence, 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].
