If 
f(x) = x − 4x2
 and h ≠ 0, find the following and simplify.
(a)    f(x + h) = 
 


(b)    
f(x + h) − f(x)
h
 = 
 

Given the function \( f(x) = x - 4x^2 \), let's solve each part step by step.

### (a) Find \( f(x + h) \):
We need to substitute \( x + h \) into the function \( f(x) = x - 4x^2 \).

\[
f(x + h) = (x + h) - 4(x + h)^2
\]

Now expand the terms:

\[
f(x + h) = (x + h) - 4(x^2 + 2xh + h^2)
\]
\[
f(x + h) = (x + h) - 4x^2 - 8xh - 4h^2
\]
\[
f(x + h) = x + h - 4x^2 - 8xh - 4h^2
\]

So, 

\[
f(x + h) = -4x^2 - 8xh - 4h^2 + x + h
\]

### (b) Find \( \frac{f(x + h) - f(x)}{h} \):

We already have \( f(x + h) \) and we know that \( f(x) = x - 4x^2 \). Now, let's compute the difference \( f(x + h) - f(x) \):

\[
f(x + h) - f(x) = (-4x^2 - 8xh - 4h^2 + x + h) - (x - 4x^2)
\]

Simplify the terms:

\[
f(x + h) - f(x) = -4x^2 - 8xh - 4h^2 + x + h - x + 4x^2
\]

\[
f(x + h) - f(x) = (-4x^2 + 4x^2) + (-x + x) - 8xh - 4h^2 + h
\]

\[
f(x + h) - f(x) = -8xh - 4h^2 + h
\]

Now, divide by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = \frac{-8xh - 4h^2 + h}{h}
\]

Simplify the expression by dividing each term by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = -8x - 4h + 1
\]

Thus, the simplified expression is:

\[
\frac{f(x + h) - f(x)}{h} = -8x - 4h + 1
\]

### Summary of results:

(a) \( f(x + h) = -4x^2 - 8xh - 4h^2 + x + h \)

(b) \( \frac{f(x + h) - f(x)}{h} = -8x - 4h + 1 \)

Would you like more details on any of these steps? Here are 5 related questions:

1. How would this change if the function were cubic instead of quadratic?
2. What does the expression \( \frac{f(x + h) - f(x)}{h} \) represent in calculus?
3. Can you simplify \( f(x + h) \) for a different function, say \( f(x) = 3x^2 - 2x \)?
4. How would the result change if \( h \) approaches zero?
5. How would you apply the same steps to a trigonometric function?

**Tip**: The difference quotient \( \frac{f(x + h) - f(x)}{h} \) is a fundamental concept in calculus and is used to find the derivative of a function.

If f(x) = x − 4x^2 and h ≠ 0, find the following and simplify. (a) f(x + h) = ? (b) (f(x + h) − f(x))/h = ?

Learn how to find and simplify the difference quotient for the quadratic function f(x) = x − 4x^2. This detailed step-by-step guide covers the algebraic manipulation of functions, difference quotient, and the foundational calculus concept of the derivative. Suitable for students in Grades 10-12.

Math Problem Statement

(b)
f(x + h) − f(x) h

Solution

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