Question
For the function \( f(x)=-6 x+4 \), evaluate and simplify the difference quotient.
Ask by Vaughan Frazier. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The difference quotient simplifies to \(-6\).
Solution
To evaluate and simplify the difference quotient for the function \( f(x) = -6x + 4 \), follow these steps:
### Difference Quotient Formula
The difference quotient is given by:
\[
\frac{f(x + h) - f(x)}{h}
\]
### Step 1: Compute \( f(x + h) \)
\[
f(x + h) = -6(x + h) + 4 = -6x - 6h + 4
\]
### Step 2: Subtract \( f(x) \) from \( f(x + h) \)
\[
f(x + h) - f(x) = (-6x - 6h + 4) - (-6x + 4)
\]
\[
= -6x - 6h + 4 + 6x - 4
\]
\[
= -6h
\]
### Step 3: Divide by \( h \)
\[
\frac{f(x + h) - f(x)}{h} = \frac{-6h}{h} = -6 \quad \text{(for } h \neq 0\text{)}
\]
### Final Answer
\[
\frac{f(x + h) - f(x)}{h} = -6
\]
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Simplify this solution Extra Insights
To evaluate the difference quotient for the function \( f(x) = -6x + 4 \), we start with the formula: \[ \frac{f(x+h) - f(x)}{h} \] First, compute \( f(x+h) \): \[ f(x+h) = -6(x+h) + 4 = -6x - 6h + 4 \] Now, substituting \( f(x) \) and \( f(x+h) \) into the difference quotient: \[ f(x+h) - f(x) = (-6x - 6h + 4) - (-6x + 4) = -6h \] Now, we can plug this into the difference quotient: \[ \frac{-6h}{h} \] Assuming \( h \neq 0 \), the \( h \) cancels out: \[ -6 \] So, the simplified difference quotient is: \[ -6 \]
