Pre-AP Calculus

Book: PRECALCULUS Mathematics for Calculus by James Stewart, Lothar Redlin, Saleem Watson.

Function Definition
A function \( f \) assigns to each element \( x \) in a set \( A \) exactly one element, denoted as \( f(x) \), in a set \( B \).

Following is not a function.

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

### Step 1: Compute \( f(x + h) \) \[ f(x + h) = 2(x + h)^2 + 3(x + h) - 1 \] \[ = 2(x^2 + 2xh + h^2) + 3x + 3h - 1 \] \[ = 2x^2 + 4xh + 2h^2 + 3x + 3h - 1 \]

### Step 2: Compute \( f(x + h) - f(x) \) \[ f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 + 3x + 3h - 1\right) - \left(2x^2 + 3x - 1\right) \] \[ = 4xh + 2h^2 + 3h \]

### Step 3: Form the Difference Quotient \[ \frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 + 3h}{h} \] \[ = \frac{h(4x + 2h + 3)}{h} \] \[ = 4x + 2h + 3 \quad \text{(for \( h \neq 0 \))} \]

Recall that the domain of a function is the set of all inputs for the function.