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. Be sure to simplify.

f left parenthesis x right parenthesis equals x squared minus 5 x plus 7

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

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

For the function \( f(x) = x^2 - 5x + 7 \), follow these steps to find the difference quotient:

### Step 1: Compute \( f(x + h) \)

Substitute \( x + h \) into \( f(x) = x^2 - 5x + 7 \):

\[
f(x + h) = (x + h)^2 - 5(x + h) + 7
\]

Expand each term:

\[
(x + h)^2 = x^2 + 2xh + h^2
\]

\[
-5(x + h) = -5x - 5h
\]

So,

\[
f(x + h) = x^2 + 2xh + h^2 - 5x - 5h + 7
\]

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

Now subtract \( f(x) = x^2 - 5x + 7 \) from \( f(x + h) \):

\[
f(x + h) - f(x) = \left(x^2 + 2xh + h^2 - 5x - 5h + 7\right) - \left(x^2 - 5x + 7\right)
\]

Simplify by canceling out the common terms:

\[
f(x + h) - f(x) = 2xh + h^2 - 5h
\]

### Step 3: Divide by \( h \)

Now, divide the expression by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2 - 5h}{h}
\]

Simplify the expression by factoring \( h \) out of the numerator:

\[
\frac{f(x + h) - f(x)}{h} = \frac{h(2x + h - 5)}{h}
\]

Since \( h \neq 0 \), we can cancel out \( h \):

\[
\frac{f(x + h) - f(x)}{h} = 2x + h - 5
\]

### Final Answer:

The simplified difference quotient is:

\[
\boxed{2x + h - 5}
\]

Do you want more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How do you find the difference quotient for a cubic function?
2. What happens to the difference quotient as \( h \to 0 \)?
3. How is the difference quotient related to the derivative?
4. How do you compute the difference quotient for a function with a radical expression?
5. Can you compute the difference quotient for \( f(x) = 3x^2 - 4x + 1 \)?

**Tip:** The difference quotient is a fundamental concept for understanding the derivative in calculus.

Find the difference quotient of f; that is, find \( \frac{f(x+h) - f(x)}{h} \), h ≠ 0, for the function f(x) = x^2 - 5x + 7. Be sure to simplify.

Find the simplified difference quotient for the quadratic function f(x) = x^2 - 5x + 7. This step-by-step guide covers key algebraic concepts, including expanding and simplifying quadratic expressions, as well as applying the difference quotient formula. Suitable for high school students in Grades 10-12.

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