Question

question 16 of 25
what is the domain of ( f(x) = 3^x )?

a. all nonnegative real numbers
b. all real numbers greater than or equal to 3
c. all real numbers
d. all real numbers greater than 3

Explanation:

Step1: Recall domain of exponential functions

Exponential functions of the form \( f(x) = a^x \) (where \( a>0, a
eq1 \)) have a domain of all real numbers because any real number \( x \) can be used as the exponent. For \( f(x) = 3^x \), \( a = 3>0 \) and \( a
eq1 \), so \( x \) can be any real number.

Step2: Evaluate options

• Option A: Nonnegative real numbers exclude negative reals, but \( 3^x \) is defined for negative \( x \) (e.g., \( 3^{-1}=\frac{1}{3} \)), so A is wrong.
• Option B: Requiring \( x\geq3 \) is incorrect as \( x \) can be less than 3 (e.g., \( x = 0 \), \( f(0)=1 \)).
• Option C: Matches the domain of exponential functions, so C is correct.
• Option D: Requiring \( x>3 \) is incorrect as \( x \) can be 3 or less (e.g., \( x = 3 \), \( f(3)=27 \); \( x = 2 \), \( f(2)=9 \)).

Answer:

C. All real numbers