If `f(x)=4x-5,g(x)=x^(2) and h(x)=(1)/(x)`, then `f(g(h(x)))` is :

Options

A. `(4)/((x-5))`
B. `(1)/((4x-5)^(2))`
C. `((x)/(4)-5)`
D. `((4)/(x^(2))-5)`

Answer

To solve the problem, we need to find the composition of the functions \( f(g(h(x))) \) where:

- \( f(x) = 4x - 5 \)
- \( g(x) = x^2 \)
- \( h(x) = \frac{1}{x} \)

Let's break this down step by step.

### Step 1: Find \( h(x) \)
Given:
\[ h(x) = \frac{1}{x} \]

### Step 2: Find \( g(h(x)) \)
Now we need to substitute \( h(x) \) into \( g(x) \):
\[ g(h(x)) = g\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \]

### Step 3: Find \( f(g(h(x))) \)
Next, we substitute \( g(h(x)) \) into \( f(x) \):
\[ f(g(h(x))) = f\left(\frac{1}{x^2}\right) = 4\left(\frac{1}{x^2}\right) - 5 \]
This simplifies to:
\[ f(g(h(x))) = \frac{4}{x^2} - 5 \]

### Final Result
Thus, the final expression for \( f(g(h(x))) \) is:
\[ f(g(h(x))) = \frac{4}{x^2} - 5 \]