If `f(x) = 1/2 (3^x + 3^(-x)), g(x) = 1/2 (3^x - 3^(-x))`, then `f(x) \ g(y)+f(y) \ g(x)=`

Options

A. f (x+y)
B. g(x+y)
C. 2f (x)
D. 2g (x)

Answer

To solve the problem, we need to evaluate the expression \( f(x) \cdot g(y) + f(y) \cdot g(x) \) where:

\[ f(x) = \frac{1}{2} (3^x + 3^{-x}) \]
\[ g(x) = \frac{1}{2} (3^x - 3^{-x}) \]

### Step 1: Write down the expressions for \( f(x) \) and \( g(y) \)

We have:
\[ f(x) = \frac{1}{2} (3^x + 3^{-x}) \]
\[ g(y) = \frac{1}{2} (3^y - 3^{-y}) \]

### Step 2: Calculate \( f(x) \cdot g(y) \)

Now we calculate \( f(x) \cdot g(y) \):
\[ f(x) \cdot g(y) = \left(\frac{1}{2} (3^x + 3^{-x})\right) \cdot \left(\frac{1}{2} (3^y - 3^{-y})\right) \]
\[ = \frac{1}{4} (3^x + 3^{-x})(3^y - 3^{-y}) \]

### Step 3: Expand the product

Now we expand the product:
\[ = \frac{1}{4} \left( 3^x \cdot 3^y - 3^x \cdot 3^{-y} + 3^{-x} \cdot 3^y - 3^{-x} \cdot 3^{-y} \right) \]
\[ = \frac{1}{4} \left( 3^{x+y} - 3^{x-y} + 3^{-x+y} - 3^{-x-y} \right) \]

### Step 4: Calculate \( f(y) \cdot g(x) \)

Next, we calculate \( f(y) \cdot g(x) \):
\[ f(y) = \frac{1}{2} (3^y + 3^{-y}) \]
\[ g(x) = \frac{1}{2} (3^x - 3^{-x}) \]
\[ f(y) \cdot g(x) = \left(\frac{1}{2} (3^y + 3^{-y})\right) \cdot \left(\frac{1}{2} (3^x - 3^{-x})\right) \]
\[ = \frac{1}{4} (3^y + 3^{-y})(3^x - 3^{-x}) \]

### Step 5: Expand this product

Now we expand this product:
\[ = \frac{1}{4} \left( 3^y \cdot 3^x - 3^y \cdot 3^{-x} + 3^{-y} \cdot 3^x - 3^{-y} \cdot 3^{-x} \right) \]
\[ = \frac{1}{4} \left( 3^{y+x} - 3^{y-x} + 3^{-y+x} - 3^{-y-x} \right) \]

### Step 6: Add \( f(x) \cdot g(y) \) and \( f(y) \cdot g(x) \)

Now we add the two results:
\[ f(x) \cdot g(y) + f(y) \cdot g(x) = \frac{1}{4} \left( 3^{x+y} - 3^{x-y} + 3^{-x+y} - 3^{-x-y} \right) + \frac{1}{4} \left( 3^{y+x} - 3^{y-x} + 3^{-y+x} - 3^{-y-x} \right) \]

### Step 7: Simplify the expression

Combining the two fractions:
\[ = \frac{1}{4} \left( 2 \cdot 3^{x+y} - (3^{x-y} + 3^{y-x}) + (3^{-x+y} + 3^{-y+x}) - 2 \cdot 3^{-x-y} \right) \]
\[ = \frac{1}{2} \cdot 3^{x+y} - \frac{1}{4} (3^{x-y} + 3^{y-x}) + \frac{1}{4} (3^{-x+y} + 3^{-y+x}) - \frac{1}{2} \cdot 3^{-x-y} \]

### Step 8: Recognize the pattern

Notice that:
\[ g(x+y) = \frac{1}{2} (3^{x+y} - 3^{-(x+y)}) \]
Thus, we can conclude:
\[ f(x) \cdot g(y) + f(y) \cdot g(x) = g(x+y) \]

### Final Result

So, the final answer is:
\[ f(x) \cdot g(y) + f(y) \cdot g(x) = g(x+y) \]