Question

identifying domain and range of exponential fun
identify the domain and range of the function.
$y = 3 \cdot 5^x$
the domain of this function is
the range of this function is

Explanation:

Step1: Determine the domain of \( y = 3\cdot5^{x} \)

For an exponential function of the form \( y = a\cdot b^{x} \) (where \( a
eq0 \) and \( b>0, b
eq1 \)), the independent variable \( x \) can take any real number value. There are no restrictions on \( x \) (like division by zero or square roots of negative numbers) here. So the domain is all real numbers.

Step2: Determine the range of \( y = 3\cdot5^{x} \)

First, consider the base function \( y = 5^{x} \). The range of \( y = 5^{x} \) is \( y>0 \) because any positive number raised to a real power is positive. Now, when we multiply this function by 3 (a positive constant), the inequality direction remains the same. So for \( y = 3\cdot5^{x} \), we have \( 3\cdot5^{x}>0 \) (since \( 5^{x}>0 \) and multiplying by 3, which is positive, keeps it positive). So the range is all positive real numbers.

Answer:

The domain of this function is all real numbers (in interval notation, \( (-\infty, \infty) \)).
The range of this function is all positive real numbers (in interval notation, \( (0, \infty) \)).