Option B:
[tex]f(x)=5^{x}[/tex] is increasing function.
Solution:
A function is increasing when the f(x) value increases as x-value increases.
Option A: [tex]f(x)=\left(\frac{1}{15}\right)^{x}[/tex]
Substitute x = 1 and x = 2 in f(x).
[tex]f(1)=\left(\frac{1}{15}\right)^{1}=0.0666[/tex]
[tex]f(2)=\left(\frac{1}{15}\right)^{2}=0.0044[/tex]
0.0666 > 0.0044
Here x is increasing but f(x) is decreasing.
So, the function is not increasing.
Option B: [tex]f(x)=5^{x}[/tex]
Substitute x = 1 and x = 2 in f(x).
[tex]f(1)=5^{1}=5[/tex]
[tex]f(2)=5^{2}=25[/tex]
5 < 25
Here f(x) is increasing as x is increasing.
So, the function is increasing.
Option C: [tex]f(x)=\left(\frac{1}{5}\right)^{x}[/tex]
Substitute x = 1 and x = 2 in f(x).
[tex]f(1)=\left(\frac{1}{5}\right)^{1}=0.2[/tex]
[tex]f(2)=\left(\frac{1}{5}\right)^{2}=0.04[/tex]
0.2 > 0.04
Here x is increasing but f(x) is decreasing.
So, the function is not increasing.
Option D: [tex]f(x)=(0.5)^{x}[/tex]
[tex]f(1)=(0.5)^{1}=0.5[/tex]
[tex]f(2)=(0.5)^{2}=0.25[/tex]
0.5 > 0.25
Here x is increasing but f(x) is decreasing.
So, the function is not increasing.
Option B is the correct answer.
Hence [tex]f(x)=5^{x}[/tex] is increasing function.
