To determine which value of [tex]\( a \)[/tex] would cause the function [tex]\( f(x) = a \left(\frac{1}{3}\right)^x \)[/tex] to stretch, we need to understand what it means for the function to stretch.
In general, a function [tex]\( g(x) = a \cdot h(x) \)[/tex] will stretch vertically if [tex]\( a \)[/tex] is greater than 1. This is because the factor [tex]\( a \)[/tex] scales the output of the function by [tex]\( a \)[/tex], making the function grow faster.
Given the options:
- 0.3
- 0.9
- 1.65
- 1.5
We need to find the values of [tex]\( a \)[/tex] that are greater than 1.
- For [tex]\( a = 0.3 \)[/tex], since [tex]\( 0.3 < 1 \)[/tex], this will compress the function rather than stretch it.
- For [tex]\( a = 0.9 \)[/tex], since [tex]\( 0.9 < 1 \)[/tex], this will also compress the function rather than stretch it.
- For [tex]\( a = 1.65 \)[/tex], since [tex]\( 1.65 > 1 \)[/tex], this will cause the function to stretch.
- For [tex]\( a = 1.5 \)[/tex], since [tex]\( 1.5 > 1 \)[/tex], this will also cause the function to stretch.
Thus, the values 1.65 and 1.5 will both cause the function to stretch. However, if we are looking for the value among the given options specifically mentioned to cause the function to stretch, we choose the first such option listed.
Therefore, the value of [tex]\( a \)[/tex] that will cause the function [tex]\( f(x) = a \left(\frac{1}{3}\right)^x \)[/tex] to stretch is:
[tex]\[ \boxed{1.65} \][/tex]
