To identify the domain and range of the function [tex]\( y = 3 \cdot 5^x \)[/tex], we need to analyze the behavior of the function and the values [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can take.
### Domain
1. The expression [tex]\( 5^x \)[/tex] represents an exponential function where the base is 5 and the exponent is [tex]\( x \)[/tex].
2. In general, exponential functions such as [tex]\( 5^x \)[/tex] are defined for all real numbers [tex]\( x \)[/tex].
3. There are no restrictions on the values [tex]\( x \)[/tex] can take, meaning [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. To find the range, consider the output values of the function, i.e., the possible values of [tex]\( y \)[/tex].
2. The term [tex]\( 5^x \)[/tex] is positive for any real number [tex]\( x \)[/tex] (since 5 raised to any power is always positive).
3. Since [tex]\( y \)[/tex] is given by [tex]\( 3 \cdot 5^x \)[/tex], and [tex]\( 5^x \)[/tex] is always positive, multiplying by 3 (a positive constant) will also yield positive values.
4. Therefore, [tex]\( y \)[/tex] will always be positive for any real number [tex]\( x \)[/tex].
5. As there is no upper limit to the values [tex]\( 5^x \)[/tex] can take, [tex]\( y \)[/tex] can grow arbitrarily large.
Hence, the range of the function [tex]\( y = 3 \cdot 5^x \)[/tex] is:
[tex]\[
\text{positive real numbers}
\][/tex]
Now we can summarize the results:
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]\[
\text{positive real numbers}
\][/tex]
