To determine the domain and range of the function [tex]\( y = 3 \cdot 5^x \)[/tex], let's analyze each aspect step by step.
### Step 1: Identify the Domain
The domain of a function is the set of all possible input values (x-values) that the function can accept.
1. The given function is [tex]\( y = 3 \cdot 5^x \)[/tex].
2. Consider any value of [tex]\( x \)[/tex]. Since there is no restriction on [tex]\( x \)[/tex], [tex]\( x \)[/tex] can be any real number.
3. There are no denominators, square roots, logarithms, or any other operations that limit [tex]\( x \)[/tex] to specific values.
Therefore, the domain of the function is:
[tex]\[
\text{All real numbers}
\][/tex]
This means [tex]\( x \)[/tex] can be any number from negative infinity to positive infinity.
### Step 2: Identify the Range
The range of a function is the set of all possible output values (y-values) that the function can produce.
1. The given function is [tex]\( y = 3 \cdot 5^x \)[/tex].
2. Analyze the exponential part of the function, [tex]\( 5^x \)[/tex].
- For [tex]\( x \to -\infty \)[/tex], [tex]\( 5^x \)[/tex] approaches 0.
- For [tex]\( x = 0 \)[/tex], [tex]\( 5^x = 1 \)[/tex].
- For [tex]\( x \to +\infty \)[/tex], [tex]\( 5^x \)[/tex] approaches infinity.
3. Since [tex]\( 5^x \)[/tex] is always positive for all [tex]\( x \)[/tex], when we multiply it by 3 (which is also positive), the result [tex]\( y = 3 \cdot 5^x \)[/tex] will always be positive.
4. No matter what [tex]\( x \)[/tex] is, [tex]\( y \)[/tex] will never be zero or negative, only positive. Thus, [tex]\( y \)[/tex] can take any positive real number.
Therefore, the range of the function is:
[tex]\[
\text{All positive real numbers}
\][/tex]
This means [tex]\( y \)[/tex] can be any number greater than 0.
### Conclusion
The domain and range of the function [tex]\( y = 3 \cdot 5^x \)[/tex] are:
- Domain: [tex]\(\text{All real numbers}\)[/tex]
- Range: [tex]\(\text{All positive real numbers}\)[/tex]
So, filling in the blanks, we have:
The domain of this function is
[tex]\(\boxed{\text{All real numbers}}\)[/tex].
The range of this function is
[tex]\(\boxed{\text{All positive real numbers}}\)[/tex].
