What are the domain and range for the exponential function f left parenthesis x right parenthesis equals a b superscript x baseline, where b is a positive real number not equal to 1 and a greater than 0?

Question

(1 point)
Responses

domain: left parenthesis negative infinity comma infinity right parenthesis; range: left parenthesis negative infinity comma 0 right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma infinity right parenthesis ; range: Image with alt text: left parenthesis negative infinity comma 0 right parenthesis

domain: left parenthesis negative infinity comma 0 right bracket; range: left parenthesis negative infinity comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma 0 right bracket ; range: Image with alt text: left parenthesis negative infinity comma infinity right parenthesis

domain: left parenthesis negative infinity comma infinity right parenthesis; range: left parenthesis 0 comma infinity right parenthesis
domain: Image with alt text: left parenthesis negative infinity comma infinity right parenthesis ; range: Image with alt text: left parenthesis 0 comma infinity right parenthesis

domain: left parenthesis 0 comma infinity right parenthesis; range: left parenthesis negative infinity comma infinity right parenthesis

Answer 1

For the exponential function \( f(x) = a b^x \), where \( b \) is a positive real number not equal to 1 and \( a > 0 \), the domain and range are as follows:

• Domain: The domain of the function is all real numbers. Therefore, the domain is \( (-\infty, \infty) \).

• Range: Since \( a > 0 \) and \( b > 0 \), the function \( f(x) = a b^x \) will always be positive, and thus the range is \( (0, \infty) \).

So the correct response is:

Domain: \( (-\infty, \infty) \); Range: \( (0, \infty) \).