To determine the domain and range of the function [tex]\( f(x) = 2 \cdot 5^x + 3 \)[/tex], let’s break down the solution step by step.
### Domain
The domain of a function is the set of all possible input values (x-values) that the function can accept.
1. Examine the function [tex]\( f(x) = 2 \cdot 5^x + 3 \)[/tex].
2. The term [tex]\( 5^x \)[/tex] represents an exponential function, which is defined for all real numbers [tex]\( x \)[/tex].
3. There are no restrictions (such as denominators that could be zero or square roots of negative numbers) in this particular function that would limit the values of [tex]\( x \)[/tex].
Therefore, the domain of [tex]\( f(x) = 2 \cdot 5^x + 3 \)[/tex] is all real numbers:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
### Range
The range of a function is the set of all possible output values (y-values) that the function can produce.
1. Again, consider the function [tex]\( f(x) = 2 \cdot 5^x + 3 \)[/tex].
2. The term [tex]\( 5^x \)[/tex] is an exponential function that is always positive and grows rapidly as [tex]\( x \)[/tex] increases.
3. The smallest value [tex]\( 5^x \)[/tex] can take is just above 0 as [tex]\( x \)[/tex] approaches negative infinity.
4. When multiplied by 2, [tex]\( 2 \cdot 5^x \)[/tex] will still be slightly above 0 and will get much larger as [tex]\( x \)[/tex] increases.
5. Adding 3 to [tex]\( 2 \cdot 5^x \)[/tex] shifts the entire function upwards by 3 units.
6. This means the smallest value [tex]\( f(x) \)[/tex] can take is slightly above 3 (when [tex]\( x \)[/tex] approaches negative infinity), and it increases without bound as [tex]\( x \)[/tex] increases.
Therefore, the range of [tex]\( f(x) = 2 \cdot 5^x + 3 \)[/tex] is:
[tex]\[ \text{Range: } (3, \infty) \][/tex]
From the given choices:
- The correct domain is [tex]\( \text{Domain: } (-\infty, \infty) \)[/tex]
- The correct range is [tex]\( \text{Range: } (3, \infty) \)[/tex]
So the correct answers are:
- Domain: [tex]\((- \infty, \infty)\)[/tex]
- Range: [tex]\((3, \infty)\)[/tex]
