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.

If the function is given by an algebraic expression and the domain is not stated explicitly, then by convention the domain of the function is the domain of the algebraic expression—that is, the set of all real numbers for which the expression is defined as a real number.

1. verbally (by a description in words)
2. algebraically (by an explicit formula)
3. visually (by a graph)
4. numerically (by a table of values)

The most important way to visualize a function is through its graph. In this section we investigate in more detail the concept of graphing functions.

\[ f(x) = \begin{cases} x^2 & \text{if } x \leq 1, \\ 2x + 1 & \text{if } x > 1. \end{cases} \]