Question
Fine the difference quote for the function \( f(x)=3 x+5 \). Simplify your answer as much as possible. \( \frac{f(x+h)-f(x)}{h}= \)
Ask by Hampton Kelly. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The difference quotient simplifies to 3.
Solution
The difference quotient is given by
\[
\frac{f(x+h)-f(x)}{h}.
\]
For the function \( f(x)=3x+5 \), we first compute \( f(x+h) \):
\[
f(x+h)=3(x+h)+5=3x+3h+5.
\]
Now, subtract \( f(x) \):
\[
f(x+h)-f(x)= (3x+3h+5)-(3x+5)=3h.
\]
Thus, the difference quotient simplifies to
\[
\frac{3h}{h}=3.
\]
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Simplify this solution Bonus Knowledge
To find the difference quotient for the function \( f(x) = 3x + 5 \), we start by calculating \( f(x+h) \): \[ f(x+h) = 3(x+h) + 5 = 3x + 3h + 5. \] Now, we substitute this into the difference quotient formula: \[ \frac{f(x+h) - f(x)}{h} = \frac{(3x + 3h + 5) - (3x + 5)}{h}. \] Simplifying the numerator: \[ = \frac{3h}{h} = 3 \quad \text{(for } h \neq 0\text{)}. \] Thus, the simplified difference quotient is \( 3 \).
