Answer :
To determine the domain and range of the function \( f(x) = 3^x + 5 \), we need to analyze the behavior of this function step-by-step.
1. Determining the Domain:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- In the given function \( f(x) = 3^x + 5 \), the term \( 3^x \) is defined for all real numbers \( x \).
- Therefore, there are no restrictions on the values of \( x \); it can take any real number.
- Hence, the domain of the function is \((-\infty, \infty)\).
2. Determining the Range:
- The range of a function is the set of all possible output values (y-values).
- Consider the term \( 3^x \):
- For \( x = 0 \), \( 3^0 = 1 \).
- For \( x > 0 \), \( 3^x \) grows exponentially and approaches infinity.
- For \( x < 0 \), \( 3^x \) approaches 0 (but is never negative and never actually reaches 0).
- Since \( 3^x \) is always positive and the smallest it can get is approaching 0 (but never 0), adding 5 to \( 3^x \) means the smallest value \( f(x) \) can approach is 5.
- As \( x \) increases, \( 3^x \) increases without bound, hence \( f(x) \) also grows without bound.
- Therefore, the function \( f(x) \) will take all values greater than 5 but never reach 5.
- Hence, the range of the function is \( (5, \infty) \).
3. Conclusion:
- Domain: \( (-\infty, \infty) \)
- Range: \( (5, \infty) \)
Thus, the correct answer is:
- domain: [tex]\( (-\infty, \infty) \)[/tex]; range: [tex]\( (5, \infty) \)[/tex]
1. Determining the Domain:
- The domain of a function is the set of all possible input values (x-values) for which the function is defined.
- In the given function \( f(x) = 3^x + 5 \), the term \( 3^x \) is defined for all real numbers \( x \).
- Therefore, there are no restrictions on the values of \( x \); it can take any real number.
- Hence, the domain of the function is \((-\infty, \infty)\).
2. Determining the Range:
- The range of a function is the set of all possible output values (y-values).
- Consider the term \( 3^x \):
- For \( x = 0 \), \( 3^0 = 1 \).
- For \( x > 0 \), \( 3^x \) grows exponentially and approaches infinity.
- For \( x < 0 \), \( 3^x \) approaches 0 (but is never negative and never actually reaches 0).
- Since \( 3^x \) is always positive and the smallest it can get is approaching 0 (but never 0), adding 5 to \( 3^x \) means the smallest value \( f(x) \) can approach is 5.
- As \( x \) increases, \( 3^x \) increases without bound, hence \( f(x) \) also grows without bound.
- Therefore, the function \( f(x) \) will take all values greater than 5 but never reach 5.
- Hence, the range of the function is \( (5, \infty) \).
3. Conclusion:
- Domain: \( (-\infty, \infty) \)
- Range: \( (5, \infty) \)
Thus, the correct answer is:
- domain: [tex]\( (-\infty, \infty) \)[/tex]; range: [tex]\( (5, \infty) \)[/tex]
