For the given​ function, find StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction

Question

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​ f(x)equals32 minus 16 x
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Part 1
StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
equals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find \(\frac{f(x+h) - f(x)}{h}\) for the function \(f(x) = 32 - 16x\), we follow these steps:

1. Calculate \(f(x+h)\): Substitute \(x + h\) into the function: \[ f(x + h) = 32 - 16(x + h) = 32 - 16x - 16h \]

2. Calculate \(f(x)\): Given \(f(x) = 32 - 16x\).

3. Find \(f(x+h) - f(x)\): Subtract \(f(x)\) from \(f(x+h)\): \[ f(x + h) - f(x) = (32 - 16x - 16h) - (32 - 16x) \] Simplifying this: \[ f(x + h) - f(x) = 32 - 16x - 16h - 32 + 16x = -16h \]

4. Divide by \(h\): Now, we calculate: \[ \frac{f(x+h) - f(x)}{h} = \frac{-16h}{h} = -16 \quad (\text{for } h \neq 0) \]

Thus, the final simplified result is: \[ \frac{f(x+h) - f(x)}{h} = -16 \]