Questions: For the function f(x)=-6x+8, evaluate and simplify the difference quotient.

For the function f(x)=-6x+8, evaluate and simplify the difference quotient.
Transcript text: For the function $f(x)=-6 x+8$, evaluate and simplify the difference quotient. $\square$
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Solution

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Solution Steps

To evaluate and simplify the difference quotient for the function \( f(x) = -6x + 8 \), we need to use the formula for the difference quotient, which is:

\[ \frac{f(x+h) - f(x)}{h} \]

  1. Substitute \( f(x) = -6x + 8 \) into the difference quotient formula.
  2. Simplify the expression by performing the necessary algebraic operations.
  3. Simplify the resulting expression to get the final form of the difference quotient.
Step 1: Define the Difference Quotient Formula

The difference quotient for a function \( f(x) \) is given by: \[ \frac{f(x+h) - f(x)}{h} \]

Step 2: Substitute the Function into the Difference Quotient

Given \( f(x) = -6x + 8 \), we substitute this into the difference quotient formula: \[ \frac{f(x+h) - f(x)}{h} = \frac{(-6(x+h) + 8) - (-6x + 8)}{h} \]

Step 3: Simplify the Expression

Simplify the numerator: \[ (-6(x+h) + 8) - (-6x + 8) = -6x - 6h + 8 + 6x - 8 = -6h \]

Thus, the difference quotient becomes: \[ \frac{-6h}{h} = -6 \]

Final Answer

\(\boxed{-6}\)

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