Sure, let's identify the domain and range of the given function [tex]\( y = 3 \cdot 5^x \)[/tex].
### Domain:
1. The function is [tex]\( y = 3 \cdot 5^x \)[/tex].
2. The expression [tex]\( 5^x \)[/tex] is an exponential function where the base is 5, a positive constant, and the exponent is [tex]\( x \)[/tex].
3. For exponential functions of the form [tex]\( a^x \)[/tex], where [tex]\( a \)[/tex] is a positive real number, the exponent [tex]\( x \)[/tex] can take any real number value.
4. Since there's no restriction on [tex]\( x \)[/tex] for an exponential function, [tex]\( x \)[/tex] can be any real number.
Thus, the domain of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is:
[tex]\[ \text{all real numbers} \][/tex]
### Range:
1. The expression [tex]\( 5^x \)[/tex] for any real number [tex]\( x \)[/tex], always produces a positive real number because any positive number raised to any power remains positive.
2. Thus, [tex]\( 5^x > 0 \)[/tex].
3. When we multiply [tex]\( 3 \)[/tex] with [tex]\( 5^x \)[/tex], we are simply scaling the positive output of [tex]\( 5^x \)[/tex] by a factor of 3.
4. Multiplying a positive constant (3) by a positive number [tex]\( 5^x \)[/tex] still results in a positive value. Therefore, [tex]\( 3 \cdot 5^x > 0 \)[/tex].
This means the function [tex]\( y = 3 \cdot 5^x \)[/tex] can take any positive real number as its value, and it never reaches zero or becomes negative.
Thus, the range of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is:
[tex]\[ \text{all positive real numbers} \][/tex]
### Final Answer:
- The domain of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is:
[tex]\[ \text{all real numbers} \][/tex]
- The range of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is:
[tex]\[ \text{all positive real numbers} \][/tex]
