Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.