Answer:
The correct option is 3, i.e., [tex]f(x)=(\frac{1}{4})^{x-2}[/tex].
Step-by-step explanation:
From the graph it is clear that as x tends to infinity, then the value of function tends to 0 and y-intercept of the function is at above (0,8).
In option 1:
[tex]f(x)=(\frac{1}{4})^{x+2}[/tex]
At x=0,
[tex]f(0)=(\frac{1}{4})^{0+2}=0.063[/tex]
The y-intercept of this function is 0.063, which is less than 8, therefore option 1 is incorrect.
In option 2:
[tex]f(x)=(\frac{1}{4})^{x}+2[/tex]
At x=0,
[tex]f(0)=(\frac{1}{4})^{0}+2=3[/tex]
The y-intercept of this function is 3, which is less than 8, therefore option 2 is incorrect.
In option 3:
[tex]f(x)=(\frac{1}{4})^{x-2}[/tex]
At x=0,
[tex]f(0)=(\frac{1}{4})^{0-2}=16[/tex]
The y-intercept of this function is 16, which is more than 8, therefore option 3 is correct.
In option 4:
[tex]f(x)=(\frac{1}{4})^{x}-2[/tex]
At x=0,
[tex]f(0)=(\frac{1}{4})^{0}-2=-1[/tex]
The y-intercept of this function is -1, which is less than 8, therefore option 4 is incorrect.
