Which statement about p(x) = - 7(4)^ is true?

Question

Ⓐ The domain is all real numbers greater than or equal to 4.
Ⓑ The range is all real numbers less than 0.
The domain is all real numbers less than or equal to 0.
⑥ The range is all real numbers.

Answer 1

Let's analyze the function \( p(x) = -7(4^x) \).

1. Domain: The expression \( 4^x \) is defined for all real numbers \( x \). Therefore, the domain of \( p(x) \) is all real numbers, not limited to any specific interval.

2. Range: The term \( 4^x \) is always positive for all real \( x \) (since an exponential function with a positive base is always positive). Thus, \( -7(4^x) \) will always be negative, as it is the product of a negative constant and a positive quantity.

   • The smallest value of \( p(x) \) approaches 0 (but never actually reaches it, since \( 4^x \) goes to infinity as \( x \) goes to infinity). Therefore, the range is all real numbers less than 0.

Now, let's evaluate the statements:

• Ⓐ The domain is all real numbers greater than or equal to 4. (False)
• Ⓑ The range is all real numbers less than 0. (True)
• The domain is all real numbers less than or equal to 0. (False)
• ⑥ The range is all real numbers. (False)

The only true statement is Ⓑ The range is all real numbers less than 0.