Key Definitions:

• Domain: The set of all possible input values (x-values) for which the function is defined. It represents all valid independent variable values.
• Range: The set of all possible output values (y-values) that the function can produce. It represents all dependent variable values.

Notation:

Domain: \( D_f = \{x \mid x \in \mathbb{R}, \text{ such that } f(x) \text{ is defined}\} \)

Range: \( R_f = \{y \mid y = f(x), x \in D_f\} \)

Finding Domain - Key Rules:

1. Polynomial Functions: Domain = All real numbers \( \mathbb{R} \)
2. Rational Functions: Exclude values that make denominator = 0
   For \( f(x) = \frac{p(x)}{q(x)} \), domain: \( q(x) \neq 0 \)
3. Square Root Functions: Expression under radical must be ≥ 0
   For \( f(x) = \sqrt{g(x)} \), domain: \( g(x) \geq 0 \)
4. Logarithmic Functions: Argument must be > 0
   For \( f(x) = \log(g(x)) \), domain: \( g(x) > 0 \)

Examples:

1. \( f(x) = x^2 + 3x - 4 \) → Domain: \( (-\infty, \infty) \)

2. \( f(x) = \frac{1}{x-3} \) → Domain: \( x \neq 3 \) or \( (-\infty, 3) \cup (3, \infty) \)

3. \( f(x) = \sqrt{x-2} \) → Domain: \( [2, \infty) \)

Definition of a Function:

A relation is a function if each input (x-value) corresponds to exactly one output (y-value).

For every \( x \) in the domain, there exists exactly one \( y \) such that \( (x, y) \) is in the relation.

Vertical Line Test:

Rule: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Passes Test: Vertical line touches graph at most once → IS a function
• Fails Test: Vertical line intersects at two or more points → NOT a function

Examples:

• Function: \( y = x^2 \) (parabola) - passes vertical line test ✓
• Function: \( y = 2x + 3 \) (line) - passes vertical line test ✓
• Not a Function: \( x^2 + y^2 = r^2 \) (circle) - fails vertical line test ✗

Function Notation:

\( f(x) \) is read as "f of x" where f is the function name and x is the input value.

How to Evaluate Functions:

Step 1: Identify the function rule \( f(x) \)

Step 2: Substitute the given input value for every occurrence of \( x \)

Step 3: Simplify using order of operations (PEMDAS)

Step 4: The result is \( f(\text{input value}) \)

Examples:

Given: \( f(x) = 3x^2 - 2x + 5 \)

Find \( f(2) \):
\( f(2) = 3(2)^2 - 2(2) + 5 = 3(4) - 4 + 5 = 12 - 4 + 5 = 13 \)

Find \( f(-1) \):
\( f(-1) = 3(-1)^2 - 2(-1) + 5 = 3(1) + 2 + 5 = 10 \)

Find \( f(a) \):
\( f(a) = 3a^2 - 2a + 5 \)

Reading Function Values from Graphs:

To find function values from a graph, use coordinate pairs \( (x, y) \) where \( y = f(x) \)

Method 1: Find f(a) - Given x, Find y

1. Locate the value \( x = a \) on the horizontal axis
2. Draw a vertical line from \( x = a \) until it intersects the graph
3. From the intersection point, draw a horizontal line to the y-axis
4. The y-value where the horizontal line meets the y-axis is \( f(a) \)

Method 2: Find x when f(x) = b - Given y, Find x

1. Locate the value \( y = b \) on the vertical axis
2. Draw a horizontal line from \( y = b \) until it intersects the graph
3. From the intersection point(s), draw vertical line(s) to the x-axis
4. The x-value(s) where vertical line(s) meet the x-axis are the solution(s)

Key Point: Every point on the graph \( (a, b) \) satisfies \( b = f(a) \)

Steps to Complete Function Tables:

1. Identify the function equation: \( y = f(x) \)
2. For each given x-value:
   • Substitute the x-value into the function
   • Calculate the corresponding y-value
   • Write the result in the table
3. For given y-values:
   • Set \( f(x) = y \)
   • Solve the equation for x
   • Write the x-value in the table
4. Plot points: Use ordered pairs \( (x, y) \) to graph the function

Example:

Function: \( f(x) = 2x + 3 \)

When \( x = 0 \): \( f(0) = 2(0) + 3 = 3 \) → Point: \( (0, 3) \)

When \( x = 1 \): \( f(1) = 2(1) + 3 = 5 \) → Point: \( (1, 5) \)

When \( x = -2 \): \( f(-2) = 2(-2) + 3 = -1 \) → Point: \( (-2, -1) \)

Matching Real-World Situations to Graphs:

Analyze the context and behavior described in word problems to identify the correct graph.

Key Characteristics to Identify:

• Increasing vs. Decreasing: Does the quantity grow or shrink?
• Linear vs. Nonlinear: Constant rate (straight line) or changing rate (curve)?
• Starting Point: What is the initial value (y-intercept)?
• Rate of Change: Is it constant, increasing, or decreasing?
• Maximum/Minimum: Are there peak or lowest values?
• Discontinuities: Are there breaks, jumps, or vertical asymptotes?

Common Scenarios:

Linear Growth: Constant speed, steady savings → Straight line with positive slope

Exponential Growth: Population growth, compound interest → Upward curving graph

Exponential Decay: Radioactive decay, cooling → Downward curving graph approaching zero

Quadratic: Projectile motion, area problems → Parabola (U-shaped or inverted U)

Using Tables to Solve Equations:

Tables help identify exact or approximate solutions to equations \( f(x) = c \) by examining input-output pairs.

Finding Exact Solutions:

1. Create or examine the function table with x and y values
2. Look for the row where \( y = f(x) \) equals the target value
3. The corresponding x-value is the exact solution
4. If \( y = c \) appears in table, then \( x \) is the solution to \( f(x) = c \)

Approximating Solutions:

1. Find two consecutive x-values where y changes from below to above the target (or vice versa)
2. The solution lies between these two x-values
3. Use linear interpolation for better approximation:
   \( x \approx x_1 + \frac{c - y_1}{y_2 - y_1}(x_2 - x_1) \)
4. Where \( y_1 < c < y_2 \) (or opposite inequality)

Example:

Solve: \( f(x) = x^2 - 3x = 0 \)

Create table:

Solutions: \( x = 0 \) and \( x = 3 \)

Definition:

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input.

Formula:

\( \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} = \frac{\Delta y}{\Delta x} \)

Interpretation:

• Geometric: Slope of the secant line connecting points \( (a, f(a)) \) and \( (b, f(b)) \)
• Positive value: Function is increasing on average
• Negative value: Function is decreasing on average
• Zero value: No net change over the interval

Steps to Calculate:

1. Identify the interval \( [a, b] \)
2. Evaluate \( f(a) \) - the function value at the left endpoint
3. Evaluate \( f(b) \) - the function value at the right endpoint
4. Calculate the change in outputs: \( \Delta y = f(b) - f(a) \)
5. Calculate the change in inputs: \( \Delta x = b - a \)
6. Divide: \( \frac{\Delta y}{\Delta x} \)

Example:

Given: \( f(x) = x^2 + 2x \) on interval \( [1, 4] \)

Step 1: \( a = 1, b = 4 \)

Step 2: \( f(1) = (1)^2 + 2(1) = 1 + 2 = 3 \)

Step 3: \( f(4) = (4)^2 + 2(4) = 16 + 8 = 24 \)

Step 4: \( \Delta y = f(4) - f(1) = 24 - 3 = 21 \)

Step 5: \( \Delta x = 4 - 1 = 3 \)

Step 6: \( \text{Average Rate of Change} = \frac{21}{3} = 7 \)

• Domain Restrictions: Denominator ≠ 0, Radicand ≥ 0, Logarithm argument > 0
• Vertical Line Test: Function if vertical line intersects graph at most once
• Function Evaluation: Substitute input for x: \( f(a) = \text{result} \)
• Reading Graphs: Point \( (x, y) \) means \( f(x) = y \)
• Average Rate of Change: \( \frac{f(b) - f(a)}{b - a} \) = Slope of secant line