The exponential function \( y = a(b)^x \) (where \( a > 0 \) and \( b > 0 \)) has the following characteristics regarding its domain and range:
Implied Domain:
- All Real Numbers: The variable \( x \) can take any real number value. This is because you can plug any real number into the function, and it will produce a valid output. Exponential functions do not have any restrictions such as divisions by zero or square roots of negative numbers, which can limit the domain of other types of functions. Therefore, the implied domain of the function is all real numbers, \( (-\infty, \infty) \).
Range:
- All Positive Numbers: The output of the function \( y = a(b)^x \) will always be positive regardless of the value of \( x \) because:
- \( a > 0 \): The factor \( a \) scales the function vertically and is always positive.
- \( (b)^x \): For any positive base \( b \), the expression \( (b)^x \) will yield positive values for any real number \( x \) (since raising a positive number to any exponent results in a positive number).
Combining these factors, when you multiply \( a \) (a positive number) by \( (b)^x \) (also always positive), the resulting output \( y \) is always positive. Therefore, the range of the function is \( (0, \infty) \), meaning all positive numbers.
In summary:
- The domain of \( y = a(b)^x \) is all real numbers due to the absence of restrictions on \( x \).
- The range is all positive numbers because the function always produces positive outputs for any real input.