If \(f(x+y)=f(x)f(y)\) and \(f(5)=4\), then \(f(10)-f(-10)\) is equal to
- 0
- 15.9375
- 3
- 14.0625
The Correct Option is B
Approach Solution - 1
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:
\(\boxed{15.9375}\)
Approach Solution -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:
\(\boxed{15.9375}\)
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