Question

identify the domain and range of each function.
y = 3·5^x
the domain of this function is
dropdown
the range of this function is
dropdown

Explanation:

Step1: Analyze the domain of the exponential function

For the function \( y = 3\cdot5^{x} \), the base of the exponential function is \( 5 \), and there are no restrictions on the input value \( x \) (such as division by zero or taking the square root of a negative number) because any real number can be used as an exponent. So the domain is all real numbers.

Step2: Analyze the range of the exponential function

We know that for any real number \( x \), \( 5^{x}>0 \) (since the exponential function with a base greater than 0 and not equal to 1 is always positive). Then, when we multiply both sides of the inequality \( 5^{x}>0 \) by 3 (a positive number), the direction of the inequality remains the same, so \( 3\cdot5^{x}>0 \). That means the value of \( y = 3\cdot5^{x} \) is always greater than 0. So the range is all positive real numbers.

Answer:

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