https://file.aiquickdraw.com/mathbot/file/172652095180730yxj76w.jpg

Let's calculate the last expression step by step based on the image you provided.

You are given the function:

\[
f(x) = 4x + 3
\]

### Step 1: Calculate \( f(a) \)
Substitute \( a \) into the function \( f(x) \):

\[
f(a) = 4a + 3
\]

### Step 2: Calculate \( f(a + h) \)
Substitute \( a + h \) into the function \( f(x) \):

\[
f(a + h) = 4(a + h) + 3 = 4a + 4h + 3
\]

### Step 3: Find \( \frac{f(a+h) - f(a)}{h} \)
Now, subtract \( f(a) \) from \( f(a + h) \) and simplify:

\[
f(a+h) - f(a) = (4a + 4h + 3) - (4a + 3) = 4h
\]

Then, divide by \( h \):

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

Thus, the final value is:

\[
\boxed{4}
\]

The result shown as 1 in the image might be a placeholder, but the actual calculation yields 4.

Would you like further explanation or details on any step?

### Relative Questions:
1. What is the geometric interpretation of \( \frac{f(a+h) - f(a)}{h} \)?
2. How would the result change if the function was quadratic, say \( f(x) = x^2 \)?
3. Can this expression be used to find the derivative of linear functions?
4. What happens to this limit when \( h \to 0 \)?
5. How does this formula connect to the definition of a derivative?

**Tip:** The expression \( \frac{f(a+h) - f(a)}{h} \) is known as the difference quotient, which plays a crucial role in defining derivatives.

Learn how to calculate the difference quotient for the linear function f(x) = 4x + 3. This problem covers key algebraic concepts including linear functions, difference quotient formulas, and their application in calculus. Suitable for high school students in Grades 10-12.

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