Question

If \(f(x+y)=f(x)f(y)\) and \(f(5)=4\), then \(f(10)-f(-10)\) is equal to

• A. 0
• B. 15.9375
• C. 3
• D. 14.0625

Answer 1

The Correct Option is B

Given:

Functional equation: \[ f(x + y) = f(x)f(y) \] and \[ f(5) = 4 \]

Step 1: Find \( f(0) \)

Let \( y = 0 \) in the functional equation: \[ f(x + 0) = f(x)f(0) \Rightarrow f(x) = f(x)f(0) \] Divide both sides by \( f(x) \neq 0 \), we get: \[ f(0) = 1 \]

Step 2: Evaluate \( f(10) \)

\[ f(10) = f(5 + 5) = f(5) \cdot f(5) = 4 \cdot 4 = 16 \]

Step 3: Evaluate \( f(-5) \)

Using the identity: \[ f(0) = f(5 + (-5)) = f(5)f(-5) \Rightarrow 1 = 4 \cdot f(-5) \] \[ \Rightarrow f(-5) = \frac{1}{4} \]

Step 4: Evaluate \( f(-10) \)

\[ f(-10) = f(-5 + (-5)) = f(-5) \cdot f(-5) = \left( \frac{1}{4} \right)^2 = \frac{1}{16} \]

Step 5: Compute the final result

\[ f(10) - f(-10) = 16 - \frac{1}{16} = \frac{256 - 1}{16} = \frac{255}{16} = \boxed{15.9375} \]

Final Answer:

\(\boxed{15.9375}\)

Answer 2

Given: \[ f(5) = 4 \]

The function resembles the form: \[ f(x) = a^x \] Using the given value: \[ a^5 = 4 \Rightarrow a = 4^{\frac{1}{5}} \]

Therefore, the general function becomes: \[ f(x) = \left(4^{\frac{1}{5}}\right)^x = 4^{\frac{x}{5}} \]

Now compute:

\[ f(10) = 4^{\frac{10}{5}} = 4^2 = 16 \] \[ f(-10) = 4^{\frac{-10}{5}} = 4^{-2} = \frac{1}{16} \]

So, \[ f(10) - f(-10) = 16 - \frac{1}{16} = \frac{256 - 1}{16} = \frac{255}{16} = 15.9375 \]

Final Answer:

\(\boxed{15.9375}\)