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Part 1

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 commaf(x+h)−f(x)h, 

h not equals 0 commah≠0, 

for the following function.

f left parenthesis x right parenthesis equals negative 7 x plus 7f(x)=−7x+7

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Part 1

StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFractionf(x+h)−f(x)hequals=enter your response here  

 ​(Simplify your​ answer.)

The difference quotient of a function \( f(x) \) is given by:

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

Given the function:

\[
f(x) = -7x + 7
\]

### Step 1: Find \( f(x+h) \)
To find \( f(x+h) \), substitute \( x+h \) into the function \( f(x) \):

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

Simplify:

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

### Step 2: Compute \( f(x+h) - f(x) \)

Now, subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = \left(-7x - 7h + 7\right) - \left(-7x + 7\right)
\]

Simplify:

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

### Step 3: Compute the difference quotient

Now, divide by \( h \):

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

Since \( h \neq 0 \), we can cancel \( h \) from the numerator and denominator:

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

Thus, the difference quotient is \( \boxed{-7} \).

Would you like more details or explanations?

### Related Questions:
1. What is the interpretation of the difference quotient?
2. How does the difference quotient relate to the derivative of the function?
3. What is the derivative of \( f(x) = -7x + 7 \)?
4. What happens to the difference quotient as \( h \) approaches 0?
5. Can the difference quotient method be used for nonlinear functions?

### Tip:
For linear functions like \( f(x) = -7x + 7 \), the difference quotient will always be the constant slope of the line, which is the coefficient of \( x \).

Find the difference quotient of f; that is, find (f(x+h)−f(x))/h, h ≠ 0, for the following function: f(x) = -7x + 7. Simplify your answer.

This page explains how to find the difference quotient of the linear function f(x) = -7x + 7. It covers the step-by-step process of computing (f(x+h)−f(x))/h and simplifying it to find the result. Suitable for students in Grades 9-11 studying Algebra and Functions.

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