Question

question 23 of 25
what is the range of $f(x) = 5^x$?

a. all positive real numbers
b. all real numbers greater than or equal to 5
c. all real numbers greater than 5
d. all real numbers

Explanation:

Step1: Recall Exponential Function Properties

The function \( f(x) = 5^x \) is an exponential function with base \( 5>1 \). For any real number \( x \), when \( x = 0 \), \( f(0)=5^0 = 1 \); when \( x>0 \), \( 5^x>1 \) and increases as \( x \) increases; when \( x<0 \), let \( x=-a \) (\( a > 0 \)), then \( 5^x=\frac{1}{5^a} \), which is a positive number less than \( 1 \) (since \( 5^a>1 \) for \( a>0 \)). So, the output of \( 5^x \) is always positive, and it can take any positive real number value (by the properties of exponential functions, the range of \( a^x \) for \( a>0,a
eq1 \) is \( (0,+\infty) \)).

Step2: Evaluate Options

• Option A: "All positive real numbers" matches the range of \( 5^x \) as we analyzed.
• Option B: The function can take values less than \( 5 \) (e.g., \( f(0) = 1<5 \)), so this is incorrect.
• Option C: Similar to B, \( f(0)=1<5 \), so incorrect.
• Option D: The function never outputs non - positive numbers (e.g., it can't be \( 0 \) or negative), so it's not all real numbers.

Answer:

A. All positive real numbers