Given the function f, of, x, equals, 4, x, squared, minus, 3*f*(*x*)=4*x*2−3, express the value of start fraction, f, of, x, plus, h, minus, f, of, x, divided by, h, end fraction*hf*(*x*+*h*)−*f*(*x*)​ in simplest form

Given the function \( f(x) = 4x^2 - 3 \), you are asked to express the difference quotient:

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

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

\[
f(x+h) = 4(x+h)^2 - 3
\]

Expand \( (x+h)^2 \):

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

Thus,

\[
f(x+h) = 4(x^2 + 2xh + h^2) - 3
\]
\[
f(x+h) = 4x^2 + 8xh + 4h^2 - 3
\]

### Step 2: Compute \( f(x+h) - f(x) \)
Now subtract \( f(x) = 4x^2 - 3 \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 - 3) - (4x^2 - 3)
\]

Simplifying:

\[
f(x+h) - f(x) = 4x^2 + 8xh + 4h^2 - 3 - 4x^2 + 3
\]
\[
f(x+h) - f(x) = 8xh + 4h^2
\]

### Step 3: Divide by \( h \)
Now, divide the result by \( h \):

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

Simplify the expression:

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

### Final Answer
The simplified form of the difference quotient is:

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

Would you like further details or have any questions? Here are five related questions for more practice:

1. How would you evaluate the difference quotient when \( h = 0 \)?
2. How does the result \( 8x + 4h \) change as \( h \to 0 \)?
3. What is the geometric meaning of the difference quotient?
4. How do you compute the derivative of \( f(x) = 4x^2 - 3 \)?
5. How does this process differ when \( f(x) \) is a linear function?

**Tip:** The difference quotient is fundamental for finding the derivative of a function. It measures the rate of change as \( h \) approaches zero!

Given the function f(x) = 4x^2 - 3, express the value of (f(x+h) - f(x)) / h in simplest form.

This page explains how to simplify the difference quotient for the function f(x) = 4x^2 - 3. The detailed step-by-step solution involves calculating f(x+h) and finding the simplest form of (f(x+h) - f(x)) / h. This problem covers key calculus concepts, including the difference quotient and the basics of limits, suitable for students in Grades 11-12.

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