To find the equation of the exponential function in the form \( f(x) = a(b)^x \), we can use the two points provided: \((-1, \frac{1}{20})\) and \( (0, \frac{1}{4})\).
The point \( (0, \frac{1}{4}) \) gives us a direct indication of the value of \( a \):
\[ f(0) = a(b)^0 = a = \frac{1}{4}. \]
Now we know that \( a = \frac{1}{4} \), and we can substitute this into our function:
\[ f(x) = \frac{1}{4}(b)^x. \]
Next, we can use the second point \((-1, \frac{1}{20})\) to find \( b \):
\[ f(-1) = \frac{1}{4}(b)^{-1} = \frac{1}{20}. \]
This can be rewritten as:
\[ \frac{1}{4} \cdot \frac{1}{b} = \frac{1}{20}. \]
To solve for \( b \), we can cross-multiply:
\[ 20 \cdot \frac{1}{4} = 1 \cdot b. \]
\[ \frac{20}{4} = b. \]
\[ b = 5. \]
Now we can substitute \( b \) back into the function:
\[ f(x) = \frac{1}{4}(5)^x. \]
Thus, the equation for the exponential function is:
\[ \boxed{f(x) = \frac{1}{4}(5)^x}. \]