Answer :
To determine the domain and range of the function [tex]\( f(x) = 3^x + 5 \)[/tex], let's break it down step-by-step.
### Domain:
1. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
2. [tex]\( f(x) = 3^x + 5 \)[/tex] is an exponential function where the base [tex]\( 3 \)[/tex] is raised to the power of [tex]\( x \)[/tex] and then 5 is added.
3. Exponential functions [tex]\( a^x \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) are defined for all real numbers [tex]\( x \)[/tex]. In this case, [tex]\( a = 3 \)[/tex].
4. Therefore, there are no restrictions on the values of [tex]\( x \)[/tex]. The domain includes all real numbers.
Hence, the domain of [tex]\( f(x) = 3^x + 5 \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range:
1. The range of a function is the set of all possible output values (y-values) that the function can take.
2. Next, consider the behavior of [tex]\( 3^x \)[/tex]. The exponential function grows rapidly as [tex]\( x \)[/tex] increases and approaches zero but never becomes zero or negative as [tex]\( x \)[/tex] decreases. Therefore, [tex]\( 3^x > 0 \)[/tex] for all real [tex]\( x \)[/tex].
3. Since [tex]\( 3^x > 0 \)[/tex], adding 5 shifts the function upwards by 5 units. Thus, [tex]\( 3^x + 5 > 5 \)[/tex] for all [tex]\( x \)[/tex].
4. That means [tex]\( f(x) \)[/tex] will always be greater than 5, but no maximum limit exists—[tex]\( f(x) \)[/tex] can grow arbitrarily large as [tex]\( x \)[/tex] increases.
Thus, the range of [tex]\( f(x) = 3^x + 5 \)[/tex] is:
[tex]\[ (5, \infty) \][/tex]
### Conclusion:
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]
Therefore, the correct multiple-choice option is:
[tex]\[ \text{domain: } (-\infty, \infty); \text{ range: } (5, \infty) \][/tex]
### Domain:
1. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
2. [tex]\( f(x) = 3^x + 5 \)[/tex] is an exponential function where the base [tex]\( 3 \)[/tex] is raised to the power of [tex]\( x \)[/tex] and then 5 is added.
3. Exponential functions [tex]\( a^x \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) are defined for all real numbers [tex]\( x \)[/tex]. In this case, [tex]\( a = 3 \)[/tex].
4. Therefore, there are no restrictions on the values of [tex]\( x \)[/tex]. The domain includes all real numbers.
Hence, the domain of [tex]\( f(x) = 3^x + 5 \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range:
1. The range of a function is the set of all possible output values (y-values) that the function can take.
2. Next, consider the behavior of [tex]\( 3^x \)[/tex]. The exponential function grows rapidly as [tex]\( x \)[/tex] increases and approaches zero but never becomes zero or negative as [tex]\( x \)[/tex] decreases. Therefore, [tex]\( 3^x > 0 \)[/tex] for all real [tex]\( x \)[/tex].
3. Since [tex]\( 3^x > 0 \)[/tex], adding 5 shifts the function upwards by 5 units. Thus, [tex]\( 3^x + 5 > 5 \)[/tex] for all [tex]\( x \)[/tex].
4. That means [tex]\( f(x) \)[/tex] will always be greater than 5, but no maximum limit exists—[tex]\( f(x) \)[/tex] can grow arbitrarily large as [tex]\( x \)[/tex] increases.
Thus, the range of [tex]\( f(x) = 3^x + 5 \)[/tex] is:
[tex]\[ (5, \infty) \][/tex]
### Conclusion:
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]
Therefore, the correct multiple-choice option is:
[tex]\[ \text{domain: } (-\infty, \infty); \text{ range: } (5, \infty) \][/tex]
