====== Pre-AP Calculus ======
Book: PRECALCULUS Mathematics for Calculus by James Stewart, Lothar Redlin, Saleem Watson.


===== Chap.2 Function =====

==== What is a Function? ====

> 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.

{{url>https://www.desmos.com/calculator/pkb2vug0cf?embed 500,500 fullscreen | This is not a function}}

=== Evaluating 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 \( h \neq 0 \))}
\]

=== The Domain of a Function ===

> 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.//

=== Four Ways to Represent a Function ===

  - verbally (by a description in words)
  - algebraically (by an explicit formula)
  - visually (by a graph)
  - numerically (by a table of values)

==== Graphs of Functions ====

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.

=== Graphing Functions ===

=== Graphing Piecewise Defined Functions ===

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

==== Increasing and Decreasing Functions; Average Rate of Change ====

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 and Decreasing Functions ===

**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  
\[ \forall x_1, x_2 \in I, \quad x_1 < x_2 \implies f(x_1) > f(x_2). \]

=== Average Rate of Change ===

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 (\( \Delta x \)).

==== Transformations of Functions ====

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.

=== Vertical Shifting ===

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.
\[
a.\quad  g(x) = x^2 + 3 \\
b.\quad  h(x) = x^2 - 2
\]

{{url>https://www.desmos.com/calculator/uoacsjlmc5?embed 500,500}}

=== Horizontal Shifting ===

**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**


=== Reflecting Graphs ===

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.

=== Vertical Stretching and Shrinking ===

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.

=== Horizontal Stretching and Shrinking ===

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.

=== Even and Odd Functions ===

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. 

==== Quadratic Functions; Maxima and Minima ====