Answer :
To determine the domain and range of the function [tex]\( g(x) = 5^{-x} \)[/tex], let's explore the properties of exponential functions closely.
### Domain:
The domain of a function consists of all the possible input values [tex]\( x \)[/tex] for which the function is defined.
1. The function [tex]\( g(x) = 5^{-x} \)[/tex] is an exponential function.
2. In exponential functions, [tex]\( 5^{-x} \)[/tex] can be evaluated for any real number [tex]\( x \)[/tex].
3. There are no restrictions such as divisions by zero or square roots of negative numbers that might otherwise limit the domain.
Thus, the domain of [tex]\( g(x) = 5^{-x} \)[/tex] is all real numbers because every real number can be substituted for [tex]\( x \)[/tex] in the exponential function without any invalid operations.
### Range:
The range of a function consists of all possible output values [tex]\( y \)[/tex] that the function can produce.
1. For [tex]\( g(x) = 5^{-x} \)[/tex], we can rewrite it as [tex]\( g(x) = \frac{1}{5^x} \)[/tex].
2. Since [tex]\( 5^x \)[/tex] is always positive for all real [tex]\( x \)[/tex], [tex]\( \frac{1}{5^x} \)[/tex] is also always positive.
3. [tex]\( 5^{-x} \)[/tex] approaches zero but never actually reaches zero as [tex]\( x \)[/tex] approaches positive infinity.
4. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 5^{-x} \)[/tex] increases without bound, but it still remains positive.
Thus, the range of [tex]\( g(x) = 5^{-x} \)[/tex] is not all real numbers because the function [tex]\( 5^{-x} \)[/tex] only produces positive values, never reaching zero or negative numbers.
### Domain:
The domain of a function consists of all the possible input values [tex]\( x \)[/tex] for which the function is defined.
1. The function [tex]\( g(x) = 5^{-x} \)[/tex] is an exponential function.
2. In exponential functions, [tex]\( 5^{-x} \)[/tex] can be evaluated for any real number [tex]\( x \)[/tex].
3. There are no restrictions such as divisions by zero or square roots of negative numbers that might otherwise limit the domain.
Thus, the domain of [tex]\( g(x) = 5^{-x} \)[/tex] is all real numbers because every real number can be substituted for [tex]\( x \)[/tex] in the exponential function without any invalid operations.
### Range:
The range of a function consists of all possible output values [tex]\( y \)[/tex] that the function can produce.
1. For [tex]\( g(x) = 5^{-x} \)[/tex], we can rewrite it as [tex]\( g(x) = \frac{1}{5^x} \)[/tex].
2. Since [tex]\( 5^x \)[/tex] is always positive for all real [tex]\( x \)[/tex], [tex]\( \frac{1}{5^x} \)[/tex] is also always positive.
3. [tex]\( 5^{-x} \)[/tex] approaches zero but never actually reaches zero as [tex]\( x \)[/tex] approaches positive infinity.
4. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 5^{-x} \)[/tex] increases without bound, but it still remains positive.
Thus, the range of [tex]\( g(x) = 5^{-x} \)[/tex] is not all real numbers because the function [tex]\( 5^{-x} \)[/tex] only produces positive values, never reaching zero or negative numbers.
