If `f(x)=(ax)/b` and `g(x)=(cx^2)/a` , then g(f(a)) equals

Options

A. `a^2/b`
B. `(a^2 c)/b`
C. `(a^2c)/b^2`
D. `(ca^3)/b^2`

Answer

To solve the problem, we need to find \( g(f(a)) \) given the functions \( f(x) = \frac{ax}{b} \) and \( g(x) = \frac{cx^2}{a} \).

### Step-by-step Solution:

1. **Find \( f(a) \)**:
\[ f(a) = \frac{a \cdot a}{b} = \frac{a^2}{b} \]

2. **Substitute \( f(a) \) into \( g(x) \)**:
Now we need to find \( g(f(a)) \), which is \( g\left(\frac{a^2}{b}\right) \).

3. **Use the definition of \( g(x) \)**:
\[ g\left(\frac{a^2}{b}\right) = \frac{c\left(\frac{a^2}{b}\right)^2}{a} \]

4. **Calculate \( \left(\frac{a^2}{b}\right)^2 \)**:
\[ \left(\frac{a^2}{b}\right)^2 = \frac{a^4}{b^2} \]

5. **Substitute back into \( g(f(a)) \)**:
\[ g\left(\frac{a^2}{b}\right) = \frac{c \cdot \frac{a^4}{b^2}}{a} \]

6. **Simplify the expression**:
\[ g\left(\frac{a^2}{b}\right) = \frac{c \cdot a^4}{b^2 \cdot a} = \frac{c \cdot a^3}{b^2} \]

Thus, the final answer is:
\[ g(f(a)) = \frac{ca^3}{b^2} \]

### Final Answer:
\[ g(f(a)) = \frac{ca^3}{b^2} \]