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Sagot :
To determine the domain and range of the function [tex]\( f(x) = 3^x + 5 \)[/tex], let's analyze each part of the function in detail.
### Domain
The domain of a function is the set of all possible input values for which the function is defined.
1. The base component of the function [tex]\( 3^x \)[/tex] is an exponential function.
2. Exponential functions are defined for all real numbers, meaning that [tex]\( x \)[/tex] can take any real value.
Thus, the domain of [tex]\( f(x) = 3^x + 5 \)[/tex] is all real numbers:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
### Range
The range of a function is the set of all possible output values (or [tex]\( f(x) \)[/tex] values).
1. Start by considering the range of the exponential function [tex]\( g(x) = 3^x \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex], [tex]\( 3^x \)[/tex] approaches [tex]\( 0 \)[/tex] (but never actually reaches 0, it only gets arbitrarily close).
- As [tex]\( x \)[/tex] increases to [tex]\( \infty \)[/tex], [tex]\( 3^x \)[/tex] grows without bounds.
Therefore, the range of [tex]\( 3^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
2. Now, consider the effect of adding 5 to [tex]\( 3^x \)[/tex], i.e. [tex]\( f(x) = 3^x + 5 \)[/tex]:
- Every value in the range of [tex]\( 3^x \)[/tex] is shifted upwards by 5 units.
- When [tex]\( 3^x \)[/tex] is close to 0, [tex]\( f(x) \)[/tex] is close to 5.
- When [tex]\( 3^x \)[/tex] becomes very large, [tex]\( f(x) \)[/tex] also becomes very large.
This means that the smallest value [tex]\( f(x) \)[/tex] can get is just above 5, and there is no upper bound.
Thus, the range of [tex]\( f(x) = 3^x + 5 \)[/tex] is:
[tex]\[ \text{Range} = (5, \infty) \][/tex]
### Conclusion
Therefore, the correct domain and range for the function [tex]\( f(x) = 3^x + 5 \)[/tex] are:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
[tex]\[ \text{Range} = (5, \infty) \][/tex]
So, the correct answer is:
Domain: [tex]\((- \infty, \infty)\)[/tex]; Range: (5, \infty)
### Domain
The domain of a function is the set of all possible input values for which the function is defined.
1. The base component of the function [tex]\( 3^x \)[/tex] is an exponential function.
2. Exponential functions are defined for all real numbers, meaning that [tex]\( x \)[/tex] can take any real value.
Thus, the domain of [tex]\( f(x) = 3^x + 5 \)[/tex] is all real numbers:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
### Range
The range of a function is the set of all possible output values (or [tex]\( f(x) \)[/tex] values).
1. Start by considering the range of the exponential function [tex]\( g(x) = 3^x \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex], [tex]\( 3^x \)[/tex] approaches [tex]\( 0 \)[/tex] (but never actually reaches 0, it only gets arbitrarily close).
- As [tex]\( x \)[/tex] increases to [tex]\( \infty \)[/tex], [tex]\( 3^x \)[/tex] grows without bounds.
Therefore, the range of [tex]\( 3^x \)[/tex] is [tex]\( (0, \infty) \)[/tex].
2. Now, consider the effect of adding 5 to [tex]\( 3^x \)[/tex], i.e. [tex]\( f(x) = 3^x + 5 \)[/tex]:
- Every value in the range of [tex]\( 3^x \)[/tex] is shifted upwards by 5 units.
- When [tex]\( 3^x \)[/tex] is close to 0, [tex]\( f(x) \)[/tex] is close to 5.
- When [tex]\( 3^x \)[/tex] becomes very large, [tex]\( f(x) \)[/tex] also becomes very large.
This means that the smallest value [tex]\( f(x) \)[/tex] can get is just above 5, and there is no upper bound.
Thus, the range of [tex]\( f(x) = 3^x + 5 \)[/tex] is:
[tex]\[ \text{Range} = (5, \infty) \][/tex]
### Conclusion
Therefore, the correct domain and range for the function [tex]\( f(x) = 3^x + 5 \)[/tex] are:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
[tex]\[ \text{Range} = (5, \infty) \][/tex]
So, the correct answer is:
Domain: [tex]\((- \infty, \infty)\)[/tex]; Range: (5, \infty)
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