Comprehensive Study Guide on Functions, Transformations, and Graph Analysis

Domain and Range of Functions

Definitions:
- Domain: The complete set of possible input values ( $x$ -values) along the horizontal axis ( $x$ -axis) for which a function is defined.
- Range: The complete set of possible output values ( $y$ -values) along the vertical axis ( $y$ -axis) that the function outputs.
Polynomial and Quadratic Functions:
- For any standard quadratic function or polynomial function without explicit restrictions, the domain is all real numbers: $(-\infty, \infty)$ .
Determining Domain and Range from Graphs:
- Domain: Read along the $x$ -axis from left to right.
- Example: A graph bounded on the left at $x = -2$ (closed circle, included) and on the right at $x = 1$ (open circle, excluded) has the domain $[-2, 1)$ .
- Square brackets $[ \text{ or } ]$ indicate included endpoints, whereas parentheses $( \text{ or } )$ indicate excluded endpoints.
- Range: Read along the $y$ -axis from the lowest point to the highest point.
- Example: A graph starting at a minimum $y$ -value of $0$ (included) and rising to a maximum $y$ -value of $4$ (included) has the range $[0, 4]$ .
Radical Functions (Square Roots):
- For a square root function to yield real values, the expression inside the radical (the radicand) must be greater than or equal to zero.
- Solving Procedure for Domain:
1. Set the entire expression under the radical sign greater than or equal to zero ( $\ge 0$ ).
2. Isolate the variable $x$ using algebraic operations.
3. Express the resulting inequality in interval notation.
- Example 1: $f(x) = \sqrt{5x - 6}$
- Set radicand non-negative: $5x - 6 \ge 0$
- Add $6$ to both sides: $5x \ge 6$
- Divide both sides by the positive coefficient $5$ : $x \ge \frac{6}{5}$
- Numerical value: $\frac{6}{5} = 1.2$ (located between $1$ and $2$ on the number line).
- Interval Notation: $\left[\frac{6}{5}, \infty\right)$
- Example 2: $f(x) = \sqrt{5x - 9}$
- Set radicand non-negative: $5x - 9 \ge 0$
- Add $9$ to both sides: $5x \ge 9$
- Divide both sides by $5$ : $x \ge \frac{9}{5}$
- Interval Notation: $\left[\frac{9}{5}, \infty\right)$
Rational Functions:
- A rational function is undefined wherever the denominator is equal to zero.
- Example: $g(x) = \frac{1}{x}$
- The function is defined for all real numbers except $x = 0$ .
- Domain in Interval Notation: $(-\infty, 0) \cup (0, \infty)$

Piecewise-Defined Functions

Definition: A piecewise-defined function uses different sub-functions (or rules) for different sub-domains of the input variable $x$ .
Evaluating and Graphing Piecewise Functions:
- To evaluate $f(x)$ at a specific value of $x$ , identify which sub-domain interval contains $x$ and substitute $x$ exclusively into that piece's equation.
- To graph linear pieces, construct a T-chart of $(x, y)$ values by picking at least two points within or at the boundary of that specific sub-domain.
Example 1: Multi-Part Function Analysis:
- Consider a piecewise function defined across three segments: $\begin{cases} f(x) = 2x + 1 & \text{if } x \le -2 \\ f(x) = -6 & \text{if } -2 < x \le 4 \\ f(x) = -x - 4 & \text{if } x > 4 \end{cases}$
- Segment 1 ( $x \le -2$ ):
- Linear rule with slope $2$ and intercept adjustment.
- At boundary $x = -2$ : $f(-2) = 2(-2) + 1 = -3$ , yielding point $(-2, -3)$ .
- Segment 2 (-2 < x \le 4):
- Constant rule $f(x) = -6$ .
- Yields an open boundary point at $(-2, -6)$ and a closed boundary point at $(4, -6)$ .
- Segment 3 (x > 4):
- At boundary $x = 4$ (excluded boundary): $f(4) = -(4) - 4 = -8$ or evaluating near boundary points like $(4, 0)$ and $(5, -1)$ depending on the specific line equation assigned.
Example 2: Continuity Analysis:
- Consider the function: $\begin{cases} f(x) = 2x - 3 & \text{if } x \le 0 \\ f(x) = x - 4 & \text{if } x > 0 \end{cases}$
- Upper piece evaluation at $x = 0$ : $2(0) - 3 = -3$ .
- Lower piece evaluation as $x \to 0^+$ : $0 - 4 = -4$ .
- Continuity: Because $-3 \neq -4$ , the graph has a jump discontinuity at $x = 0$ .
- Domain: Combining $(-\infty, 0]$ and $(0, \infty)$ gives all real numbers: $(-\infty, \infty)$ .
Finding the Equation of a Line Segment from Points:
- Given points on a linear piece: $(-5, 4)$ and $(-3, -1)$ .
- Step 1: Calculate Slope $m$ : $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 4}{-3 - (-5)} = \frac{-5}{2} = -\frac{5}{2}$
- Step 2: Apply Point-Slope Form $y - y_1 = m(x - x_1)$ using point $(-3, -1)$ : $y - (-1) = -\frac{5}{2}(x - (-3))$ $y + 1 = -\frac{5}{2}(x + 3)$ $y + 1 = -\frac{5}{2}x - \frac{15}{2}$ $y = -\frac{5}{2}x - \frac{17}{2}$

Modeling and Applied Functions

Guided Tour Fee Structure:
- A tour company calculates charges based on participant group size $n$ :
- Groups of $1$ to $14$ people (0 < n < 15 where $n$ is an integer): Charged at a rate of $\$3$ per person.
- Groups of $15$ or more people ( $n \ge 15$ ): Charged a flat fixed fee of $\$45$ .
- Cost Function Formulation: $C(n) = \begin{cases} 3n & \text{if } 0 < n < 15 \\ 45 & \text{if } n \ge 15 \end{cases}$
- Calculations:
- $1$ person: $3(1) = \$3$
- $2$ persons: $3(2) = \$6$
- $3$ persons: $3(3) = \$9$
- $4$ persons: $3(4) = \$12$
- $n$ persons ( $1 \le n \le 14$ ): $3n$
Population Growth Indexing:
- Population data indexed to a base year where year $1990$ corresponds to $t = 0$ .
- Year $1993$ corresponds to $t = 3$ .
- If the population in $1993$ is $3059$ thousand, the data point is represented as $(3, 3059)$ or $f(3) = 3059$ .

Composition and Decomposition of Functions

Composition Definition and Notation:
- The composition of functions $f$ and $g$ is denoted by $(f \circ g)(x)$ and defined as $f(g(x))$ .
- The inner function $g(x)$ is evaluated first, and its output becomes the direct input for the outer function $f(x)$ .
Evaluating Composite Functions:
- For $f(x) = \frac{1}{x - 4}$ :
- Evaluating $f(3)$ : $f(3) = \frac{1}{3 - 4} = \frac{1}{-1} = -1$
Decomposing Composite Functions:
- Decomposition requires identifying an inner function $g(x)$ and an outer function $f(x)$ such that $h(x) = f(g(x))$ .
- Example 1: Decompose $h(x) = (x + 2)^9$
- Inner function: $g(x) = x + 2$
- Outer function: $f(x) = x^9$
- Verification: $f(g(x)) = f(x + 2) = (x + 2)^9$
- Example 2: Decompose $h(x) = \frac{1}{x - 4}$
- Inner function: $g(x) = x - 4$
- Outer function: $f(x) = \frac{1}{x}$
- Verification: $f(g(x)) = f(x - 4) = \frac{1}{x - 4}$

Function Transformations

Parent Functions:
- Absolute Value: $f(x) = |x|$ (V-shaped graph centered at $(0,0)$ ).
- Square Root: $f(x) = \sqrt{x}$ (starts at $(0,0)$ and extends rightward in quadrant I).
- Quadratic: $f(x) = x^2$ (U-shaped parabola centered at $(0,0)$ ).
Summary of Transformation Rules:
- Vertical Shifts: $y = f(x) + k$ shifts UP by $k$ units; $y = f(x) - k$ shifts DOWN by $k$ units.
- Horizontal Shifts: $y = f(x - h)$ shifts RIGHT by $h$ units; $y = f(x + h)$ shifts LEFT by $h$ units.
- Reflections:
- $y = -f(x)$ reflects across the $x$ -axis (negates output $y$ -values).
- $y = f(-x)$ reflects across the $y$ -axis (negates input $x$ -values).
- Vertical Stretching and Compression:
- $y = a \cdot f(x)$ vertically stretches by factor $a$ if a > 1, and vertically compresses if 0 < a < 1$.\n\n- **Comprehensive Transformation Step-by-Step Examples**:\n - *Example 1*: y = -\sqrt{x + 1} - 4\n 1. Start with parent function f(x) = \sqrt{x}.\n 2. Horizontal shift LEFT by 1 $unit:$ y = \sqrt{x + 1}.\n 3. Reflection across the x $-axis:$ y = -\sqrt{x + 1}.\n 4. Vertical shift DOWN by 4 $units:$ y = -\sqrt{x + 1} - 4.\n - **Key Points T-Chart**:\n - Set x = -1 $:$ y = -\sqrt{-1 + 1} - 4 = -\sqrt{0} - 4 = -4 \implies (-1, -4)\n - Set x = 3 $:$ y = -\sqrt{3 + 1} - 4 = -\sqrt{4} - 4 = -2 - 4 = -6 \implies (3, -6)\n - Set x = 8 $:$ y = -\sqrt{8 + 1} - 4 = -\sqrt{9} - 4 = -3 - 4 = -7 \implies (8, -7)\n - *Example 2*: y = \sqrt{-x + 4}\n 1. Factor out the negative inside the radical: y = \sqrt{-(x - 4)}.\n 2. Reflect across the y $-axis due to$ -x.\n 3. Shift RIGHT by 4 $units (since$ h = 4 $inside$ -(x - 4)).\n - *Example 3*: g(x) = 4|x + 3|\n 1. Start with parent function f(x) = |x|.\n 2. Shift LEFT by 3 $units:$ |x + 3|.\n 3. Vertically stretch by a factor of 4$.
- Example 4: $f(x) = |x - 2| - 3$
1. Start with parent function $f(x) = |x|$ .
2. Shift RIGHT by $2$ units.
3. Shift DOWN by $3$ units.

Graph Behaviors: Increasing, Decreasing, and Constant Intervals

Notation Rules:
- All intervals describing where a function is increasing, decreasing, or constant MUST be written using open interval notation (parentheses only, no closed brackets).
Definitions:
- Increasing: A function is increasing on an interval if f(x_2) > f(x_1) whenever x_2 > x_1 (graph goes upward from left to right).
- Decreasing: A function is decreasing on an interval if f(x_2) < f(x_1) whenever x_2 > x_1 (graph goes downward from left to right).
- Constant: A function is constant on an interval if $f(x_1) = f(x_2)$ for all points (horizontal line, slope $m = 0$ ).
Interval Analysis Example:
- Constant interval: Horizontal segment between $x = -6$ and $x = -1$ is written as $(-6, -1)$ .
- Increasing intervals: A graph rising between $x = -1$ and $x = 0$ , and rising again between $x = 3$ and $x = 5$ , is written as: $(-1, 0) \cup (3, 5)$

Even and Odd Functions (Symmetry)

Even Functions:
- Algebraic Test: $f(-x) = f(x)$ for every $x$ in the domain.
- Symmetry: Symmetric with respect to the $y$ -axis.
- Example: $f(x) = x^2 + 2$
- Test: $f(-x) = (-x)^2 + 2 = x^2 + 2 = f(x)$ .
- Conclusion: Function is Even.
Odd Functions:
- Algebraic Test: $f(-x) = -f(x)$ for every $x$ in the domain.
- Symmetry: Symmetric with respect to the origin.
- Example: $g(x) = \frac{1}{x} + 2x$
- Test: $g(-x) = \frac{1}{-x} + 2(-x) = -\frac{1}{x} - 2x = -\left(\frac{1}{x} + 2x\right) = -g(x)$
- Conclusion: Function is Odd.

Intercepts and One-to-One Functions

Finding Intercepts:
- $x$ -intercept(s): Set $y = 0$ (or $f(x) = 0$ ) and solve the equation for $x$ .
- $y$ -intercept: Set $x = 0$ and evaluate $f(0)$ to solve for $y$ .
One-to-One Functions:
- Definition: A function is one-to-one (1-to-1) if every unique output $y$ corresponds to exactly one unique input $x$ .
- Horizontal Line Test (HLT):
- A function is one-to-one if and only if no horizontal line intersects its graph more than once.
- Linear Functions (with non-zero slope): Pass HLT $\implies$ One-to-One.
- Quadratic Functions ( $y = x^2$ ): Fail HLT (horizontal lines intersect twice) $\implies$ NOT One-to-One.
- Absolute Value Functions ( $y = |x|$ ): Fail HLT $\implies$ NOT One-to-One.