Question

For the given​ function, find StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
.
​ 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:
\[
\frac{f(x+h) - f(x)}{h} = -16
\]