What is the end behavior of the graph of the exponential function f left parenthesis x right parenthesis equals b superscript x baseline when 0 less than b less than 1?

Question

(1 point)
Responses

f left parenthesis x right parenthesis right arrow infinity as x right arrow infinity comma f left parenthesis x right parenthesis right arrow 0 as x right arrow negative infinity
Image with alt text: f left parenthesis x right parenthesis right arrow infinity as x right arrow infinity comma f left parenthesis x right parenthesis right arrow 0 as x right arrow negative infinity

f left parenthesis x right parenthesis right arrow 0 as x right arrow infinity comma f left parenthesis x right parenthesis right arrow infinity as x right arrow negative infinity
Image with alt text: f left parenthesis x right parenthesis right arrow 0 as x right arrow infinity comma f left parenthesis x right parenthesis right arrow infinity as x right arrow negative infinity

f left parenthesis x right parenthesis right arrow infinity as x right arrow infinity comma f left parenthesis x right parenthesis right arrow negative infinity as x right arrow 0
Image with alt text: f left parenthesis x right parenthesis right arrow infinity as x right arrow infinity comma f left parenthesis x right parenthesis right arrow negative infinity as x right arrow 0

f left parenthesis x right parenthesis right arrow infinity as x right arrow 0 comma f left parenthesis x right parenthesis right arrow infinity as x right arrow negative infinity

Answer 1

For the exponential function \( f(x) = b^x \) where \( 0 < b < 1 \), the end behavior of the graph is as follows:

1. As \( x \) approaches infinity (\( x \to \infty \)), the function \( f(x) = b^x \) approaches 0. This is because any positive number less than 1 raised to a large power (infinity) gets closer and closer to zero.

2. As \( x \) approaches negative infinity (\( x \to -\infty \)), the function \( f(x) = b^x \) approaches infinity. This is because raising a fraction (between 0 and 1) to a negative power results in a value that grows larger and larger (since it is equivalent to taking the reciprocal of that fraction raised to a positive power).

Thus, the end behavior of the function is:

• \( f(x) \to 0 \) as \( x \to \infty \)
• \( f(x) \to \infty \) as \( x \to -\infty \)

Therefore, the correct response is:

f left parenthesis x right parenthesis right arrow 0 as x right arrow infinity, f left parenthesis x right parenthesis right arrow infinity as x right arrow negative infinity.