f(x) = `(x (x-p))/(q - p) + (x (x - q))/(p - q)` , p`ne`q. What is the value of f (p) + f(q) ?

Options

A. f (p - q)
B. f(p + q)
C. f (p (p + q))
D. f (q (p - q))

Answer

To solve the problem, we need to find the value of \( f(p) + f(q) \) for the function defined as:

\[ f(x) = \frac{x(x - p)}{q - p} + \frac{x(x - q)}{p - q} \]

### Step 1: Substitute \( x = p \) into \( f(x) \)

First, we calculate \( f(p) \):

\[ f(p) = \frac{p(p - p)}{q - p} + \frac{p(p - q)}{p - q} \]

The first term becomes zero because \( p - p = 0 \):

\[ f(p) = 0 + \frac{p(p - q)}{p - q} \]

Since \( p \neq q \), we can simplify the second term:

\[ f(p) = p \]

### Step 2: Substitute \( x = q \) into \( f(x) \)

Next, we calculate \( f(q) \):

\[ f(q) = \frac{q(q - p)}{q - p} + \frac{q(q - q)}{p - q} \]

The second term becomes zero because \( q - q = 0 \):

\[ f(q) = \frac{q(q - p)}{q - p} + 0 \]

Again, since \( p \neq q \), we can simplify the first term:

\[ f(q) = q \]

### Step 3: Add \( f(p) \) and \( f(q) \)

Now, we can find \( f(p) + f(q) \):

\[ f(p) + f(q) = p + q \]

### Conclusion

Thus, the final result is:

\[ f(p) + f(q) = p + q \]

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