To identify the domain and range of the function [tex]\( y = 3 \cdot 5^x \)[/tex], let's break it down step by step:
### Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
1. The function [tex]\( y = 3 \cdot 5^x \)[/tex] includes an exponential expression [tex]\( 5^x \)[/tex].
2. The base of an exponential function can handle any real number exponent without any restrictions or undefined behavior.
Thus, there are no restrictions on [tex]\( x \)[/tex] in the function [tex]\( y = 3 \cdot 5^x \)[/tex]. Therefore, the domain is all real numbers.
Domain: all real numbers
### Range:
The range of a function is the set of all possible output values (y-values).
1. Consider the term [tex]\( 5^x \)[/tex]. For any real number [tex]\( x \)[/tex]:
- If [tex]\( x \)[/tex] is positive, [tex]\( 5^x \)[/tex] is a large positive number.
- If [tex]\( x \)[/tex] is negative, [tex]\( 5^x \)[/tex] is a small positive number because any positive number raised to a negative exponent is a positive fraction.
- If [tex]\( x \)[/tex] is 0, [tex]\( 5^0 = 1 \)[/tex], which is still a positive number.
2. Since [tex]\( 5^x \)[/tex] is always a positive number for any real [tex]\( x \)[/tex], multiplying it by 3 (a positive constant) still results in a positive number.
So, the function [tex]\( y = 3 \cdot 5^x \)[/tex] will always output positive values and can never be zero or negative.
Range: all positive real numbers
