Let /be a function such that
`3f(x) + 22f(1/x) = 7x, AA x ne 0`, then for each `x ne 0`, `f(x)` is equal to

Options

A. `7/5(3x-2/x)`
B. `7/5(2x + 3/x)`
C. `1/5(2x + 3/x)`
D. `2/5 (3x-1/x)`

Answer

To solve the equation \(3f(x) + 22f\left(\frac{1}{x}\right) = 7x\) for \(f(x)\), we will follow these steps:

### Step 1: Write down the original equation
We start with the equation given in the problem:
\[ 3f(x) + 22f\left(\frac{1}{x}\right) = 7x \]

### Step 2: Substitute \(x\) with \(\frac{1}{x}\)
Next, we substitute \(x\) with \(\frac{1}{x}\) in the original equation:
\[ 3f\left(\frac{1}{x}\right) + 22f(x) = 7\left(\frac{1}{x}\right) \]
This simplifies to:
\[ 3f\left(\frac{1}{x}\right) + 22f(x) = \frac{7}{x} \]

### Step 3: Label the equations
Now we have two equations:
1. \(3f(x) + 22f\left(\frac{1}{x}\right) = 7x\) (Equation 1)
2. \(3f\left(\frac{1}{x}\right) + 22f(x) = \frac{7}{x}\) (Equation 2)

### Step 4: Solve for \(f\left(\frac{1}{x}\right)\) in terms of \(f(x)\)
From Equation 1, we can express \(f\left(\frac{1}{x}\right)\):
\[ 22f\left(\frac{1}{x}\right) = 7x - 3f(x) \]
\[ f\left(\frac{1}{x}\right) = \frac{7x - 3f(x)}{22} \]

### Step 5: Substitute \(f\left(\frac{1}{x}\right)\) into Equation 2
Now, substitute \(f\left(\frac{1}{x}\right)\) from the previous step into Equation 2:
\[ 3\left(\frac{7x - 3f(x)}{22}\right) + 22f(x) = \frac{7}{x} \]
Multiply through by 22 to eliminate the fraction:
\[ 3(7x - 3f(x)) + 22 \cdot 22f(x) = \frac{154}{x} \]
This simplifies to:
\[ 21x - 9f(x) + 484f(x) = \frac{154}{x} \]
Combine like terms:
\[ 21x + 475f(x) = \frac{154}{x} \]

### Step 6: Solve for \(f(x)\)
Now, isolate \(f(x)\):
\[ 475f(x) = \frac{154}{x} - 21x \]
\[ f(x) = \frac{\frac{154}{x} - 21x}{475} \]
To simplify further:
\[ f(x) = \frac{154 - 21x^2}{475x} \]

### Final Result
Thus, the function \(f(x)\) is given by:
\[ f(x) = \frac{154 - 21x^2}{475x} \]