Question 9 For the function $$ f(x)=2 x+1 $$, evaluate and simplify the difference quotient. Question Help: Video Message instructor Check Answer

Question

UpStudy AI · Accepted Answer

1. Write the difference quotient formula:
   
   $$
   \frac{f(x+h)-f(x)}{h}
   $$

2. Find $$ f(x+h) $$ for the function $$ f(x)=2x+1 $$:

$$
   f(x+h)=2(x+h)+1=2x+2h+1
   $$

3. Substitute $$ f(x+h) $$ and $$ f(x) $$ into the difference quotient:

$$
   \frac{f(x+h)-f(x)}{h}=\frac{(2x+2h+1)-(2x+1)}{h}
   $$

4. Simplify the numerator:

$$
   (2x+2h+1)-(2x+1)=2x+2h+1-2x-1=2h
   $$

5. Divide by $$ h $$:

$$
   \frac{2h}{h}=2 \quad (\text{provided } h\neq0)
   $$

The difference quotient simplifies to $$ 2 $$.
The difference quotient simplifies to 2.

Question 9
For the function , evaluate and simplify the difference quotient.
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