Find the difference quotient of​ f; that​ is, find StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction comma
h not equals 0 comma for the following function.
f left parenthesis x right parenthesis equals 7 x plus 1

To find the **difference quotient** for the function \( f(x) = 7x + 1 \), we use the formula:

\[
\frac{f(x+h) - f(x)}{h}, \quad h \neq 0
\]

### Step 1: Find \( f(x+h) \)
Start by substituting \( x+h \) into the function \( f(x) \):

\[
f(x+h) = 7(x+h) + 1
\]

Simplify:

\[
f(x+h) = 7x + 7h + 1
\]

### Step 2: Substitute into the Difference Quotient Formula
Now, use the difference quotient formula:

\[
\frac{f(x+h) - f(x)}{h} = \frac{(7x + 7h + 1) - (7x + 1)}{h}
\]

### Step 3: Simplify the Expression
Simplify the numerator:

\[
(7x + 7h + 1) - (7x + 1) = 7x + 7h + 1 - 7x - 1 = 7h
\]

Now, the difference quotient becomes:

\[
\frac{7h}{h}, \quad h \neq 0
\]

### Step 4: Cancel \( h \)
Cancel out \( h \) from the numerator and denominator:

\[
\frac{7h}{h} = 7
\]

Thus, the difference quotient is:

\[
7
\]

### Final Answer:
The difference quotient of \( f(x) = 7x + 1 \) is \( 7 \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the meaning of the difference quotient in calculus?
2. How would the result change if \( f(x) \) was a quadratic function?
3. Can you find the derivative from the difference quotient?
4. How does the difference quotient relate to the concept of a secant line?
5. What happens if you apply the difference quotient to \( f(x) = 0 \)?

**Tip:** The difference quotient is the foundation for understanding derivatives, which measure rates of change in calculus.

Find the difference quotient of​ f; that​ is, find StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction comma h not equals 0 comma for the following function. f left parenthesis x right parenthesis equals 7 x plus 1

Explore the difference quotient for the linear function f(x) = 7x + 1. This step-by-step solution demonstrates the use of the difference quotient formula and provides a clear interpretation of the result. Suitable for high school students in Grades 9-12.

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