To determine which function represents exponential growth, we need to identify the standard form of an exponential growth function, which is typically written as [tex]\( y = a \cdot b^x \)[/tex] where [tex]\( a > 0 \)[/tex] and [tex]\( b > 1 \)[/tex].
Let's examine each of the given functions one by one:
1. [tex]\( y = 14 x^{195} \)[/tex]
- This is a polynomial function, not an exponential function. The variable [tex]\( x \)[/tex] is raised to a power, which is typical for polynomial functions, not exponential growth.
2. [tex]\( y = 14(1.95)^x \)[/tex]
- This function fits the form [tex]\( y = a \cdot b^x \)[/tex] with [tex]\( a = 14 \)[/tex] and [tex]\( b = 1.95 \)[/tex]. Here, [tex]\( a \)[/tex] is greater than 0 and [tex]\( b \)[/tex] is greater than 1, which are the conditions for exponential growth.
3. [tex]\( y = \frac{14}{1.95^x} \)[/tex]
- This can be rewritten as [tex]\( y = 14 \cdot (1.95^{-x}) \)[/tex]. The base [tex]\( 1.95 \)[/tex] is raised to the power of [tex]\(-x\)[/tex], effectively turning it into an exponential decay function since [tex]\( 1.95^{-1} < 1 \)[/tex].
4. [tex]\( y = 14(0.95)^x \)[/tex]
- This function fits the form [tex]\( y = a \cdot b^x \)[/tex] with [tex]\( a = 14 \)[/tex] and [tex]\( b = 0.95 \)[/tex]. Here, although [tex]\( a > 0 \)[/tex], the base [tex]\( b = 0.95 \)[/tex] is less than 1, which represents exponential decay, not growth.
Based on the analysis, the function that represents exponential growth is:
[tex]\[ y = 14(1.95)^x \][/tex]
Hence, the correct answer is:
[tex]\[ y = 14(1.95)^x \][/tex]
This corresponds to the second function in the list provided.
