To determine whether the function [tex]\( y = 9^{0.9x} \)[/tex] represents exponential growth or decay, we need to analyze the base of the exponent and the exponent itself.
1. Examine the Base:
- The base of the exponential function here is 9.
- Since 9 is greater than 1 (i.e., 9 > 1), it indicates that the function tends to increase as [tex]\(x\)[/tex] increases, provided the exponent is positive.
2. Examine the Exponent:
- The exponent in the function is [tex]\( 0.9x \)[/tex].
- Notice that [tex]\( 0.9 \)[/tex] is a positive number (i.e., 0.9 > 0).
- For [tex]\( x > 0 \)[/tex], the value of [tex]\( 0.9x \)[/tex] will always be positive as well.
3. Combining Both Observations:
- Since the base (9) is greater than 1 and the exponent ([tex]\( 0.9x \)[/tex]) is positive for [tex]\( x > 0 \)[/tex], the overall function [tex]\( y = 9^{0.9x} \)[/tex] will increase as [tex]\( x \)[/tex] increases.
- This behavior — where the function value increases as [tex]\( x \)[/tex] increases — is characteristic of exponential growth.
Therefore, [tex]\( y = 9^{0.9x} \)[/tex] is an example of an exponential growth function.
So, the correct classification of the function is:
[tex]\[
\boxed{\text{Growth}}
\][/tex]
