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.

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

Functions are often used to model changing quantities. In this section we learn how to determine if a function is increasing or decreasing, and how to find the rate at which its values change as the variable changes.

Increasing Function: A function \( f \) is increasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) \leq f(x_2). \]

Strictly Increasing Function: A function \( f \) is strictly increasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) < f(x_2). \]

Decreasing Function: A function \( f \) is decreasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) \geq f(x_2). \]

Strictly Decreasing Function: A function \( f \) is strictly decreasing on an interval \( I \) if \[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) > f(x_2). \]

Definition: For a function \( f(x) \), the average rate of change over the interval \([a, b]\) is given by:

\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]

This formula calculates the change in the function's value (\( \Delta y \)) divided by the change in the input (\( \Delta x \)).

In this section we study how certain transformations of a function affect its graph. This will give us a better understanding of how to graph functions. The transformations we study are shifting, reflecting, and stretching.

Adding a constant to a function shifts its graph vertically: upward if the constant is positive and downward if it is negative.

Use the graph \( f(x) = x^2 \) of to sketch the graph of each function. \[ a.\quad g(x) = x^2 + 3 \\ b.\quad h(x) = x^2 - 2 \]

Shift Right by \( h \) Units

\[ y = f(x - h) \]

Example: \( f(x) = x^2 \) shifted right by 2:

\[ y = (x - 2)^2 \]

Shift Left by \( h \) Units

\[ y = f(x + h) \]

Example: \( f(x) = \sqrt{x} \) shifted left by 3:

\[ y = \sqrt{x + 3} \]

Key Rule

\( -h \) inside \( f(x - h) \) → shifts right

\( +h \) inside \( f(x + h) \) → shifts left

To graph \( y = -f(x) \), reflect the graph of \( y = f(x) \) in the x-axis. To graph \( y = f(-x) \), reflect the graph of \( y = f(x) \) in the y-axis.

To graph \( y = cf(x) \),

If \( c > 1 \), stretch the graph of \( y = f(x) \) vertically by a factor of c.

If \( 0 < c < 1 \), shrink the graph of \( y = f(x) \) vertically by a factor of c.

To graph \( y = f(cx) \),

If \( c > 1 \), stretch the graph of \( y = f(x) \) vertically by a factor of 1/c.

If \( 0 < c < 1 \), shrink the graph of \( y = f(x) \) vertically by a factor of 1/c.

If a function \( f \) satisfies \( f(x) = f(-x) \) for every number \( x \) in its domain, then \( f \) is called an even function.

If \( f \) satisfies \( f(x) = -f(-x) \) for every number \( x \) in its domain, then \( f \) is called an odd function.