Let's analyze the given function [tex]\( g(x) = 4 \cdot (0.15)^x \)[/tex].
### Type of the Function
The function [tex]\( g(x) = 4 \cdot (0.15)^x \)[/tex] is an exponential function. In general, exponential functions have the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is a constant and [tex]\( b \)[/tex] is the base of the exponential. Here, [tex]\( a = 4 \)[/tex] and [tex]\( b = 0.15 \)[/tex].
### Domain of the Function
The domain of an exponential function is all real numbers. This is because you can substitute any real number for [tex]\( x \)[/tex], and the function value will be defined.
Domain: All real numbers
### Range of the Function
To find the range of the function, let's consider the behavior of the function as [tex]\( x \)[/tex] changes:
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]):
[tex]\[ (0.15)^x \to 0 \][/tex]
However, it never actually reaches 0. Since it is multiplied by 4, [tex]\( g(x) \)[/tex] will approach 0 from the positive side but never become negative or zero.
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
[tex]\[ (0.15)^x \to \infty \][/tex]
Multiplying by 4 only scales this up.
Therefore, the function [tex]\( g(x) \)[/tex] will take on positive values, but will not include 0. It ranges from just above 0 to positive infinity.
Range: [tex]\((0, \infty)\)[/tex]
Summary:
- Type: Exponential function
- Domain: All real numbers
- Range: [tex]\((0, \infty)\)[/tex]
