Functions, Graphs, Domain, and Range Study Guide

Independent and Dependent Variables

• Definitions: For a given input $x$, the output of a function $f$ is called the value of $f$ at $x$ or the image of $x$ under $f$.

• Functional Equation: Function output is often denoted by a single letter, such as $y$, and written as the equation:     $y = f(x)$

• Variables: In the equation $y = f(x)$, the variable $x$ is referred to as the independent variable (or argument), and the variable $y$ is referred to as the dependent variable.

• Conceptual Meaning: The terminology suggests that $x$ is free to vary, but once a specific value is assigned to $x$, a corresponding value of $y$ is determined.

• Real-valued Functions: If the independent and dependent variables are real numbers, $f$ is called a real-valued function of a real variable.

• Example 1 (Table 0.1.2 Data): A functional relationship $y = f(x)$ is described by the following associations:     * $f(0) = 3$: The function associates $y = 3$ with $x = 0$.     * $f(1) = 4$: The function associates $y = 4$ with $x = 1$.     * $f(2) = -1$: The function associates $y = -1$ with $x = 2$.     * $f(3) = 6$: The function associates $y = 6$ with $x = 3$.

• Example 2 (Formulaic Evaluation): Given the function defined by $f(x) = 3x^2 - 4x + 2$, specific outputs are calculated by substituting the input $x$ into the formula:     * $f(0) = 3(0)^2 - 4(0) + 2 = 2$     * $f(-1.7) = 3(-1.7)^2 - 4(-1.7) + 2 = 17.47$     * $f(\sqrt{2}) = 3(\sqrt{2})^2 - 4\sqrt{2} + 2 = 6 - 4\sqrt{2} + 2 = 8 - 4\sqrt{2}$

Graphs of Functions

• Definition: The graph of a real-valued function $f$ in the $xy$-plane is defined as the graph of the equation $y = f(x)$.

• Coordinates: Each point on the graph is of the form $(x, f(x))$, where the $y$-coordinate represents the value of the function at the corresponding $x$-coordinate.

• Common Functions and their Graphs:     * $y = x$: A straight line passing through the origin.     * $y = x^2$: A parabola opening upwards from the origin.     * $y = x^3$: A cubic curve passing through the origin, appearing in the first and third quadrants.     * $y = \frac{1}{x}$: A hyperbola with branches in the first and third quadrants; undefined at $x = 0$.     * $y = \sqrt{x}$: A curve starting at the origin and extending into the first quadrant. It is imaginary for $x < 0$, so there are no points where the $x$-coordinate is negative.     * $y = \sqrt[3]{x}$: A curve extending through the origin into the first and third quadrants.

• Zeros and Intercepts: The values of $x$ for which $f(x) = 0$ are known as the zeros of $f$, the roots of the equation $f(x) = 0$, or the $x$-intercepts of the graph.

• Graph Extents: Unless otherwise indicated, it is understood that graphs extend indefinitely beyond the boundaries of the displayed figures.

The Vertical Line Test

• Criterion 0.1.3: A curve in the $xy$-plane is the graph of some function $f$ if and only if no vertical line intersects the curve more than once.

• Logic: A function cannot assign two different values to a single input $a$. If a vertical line at $x = a$ cuts a curve at $(a, b)$ and $(a, c)$, then $f(a)$ would have to be both $b$ and $c$, which is impossible for a function.

• Example 3 (The Circle): The equation $x^2 + y^2 = 25$ represents a circle of radius $5$ centered at the origin. Because vertical lines can intersect this circle twice, the equation does not define $y$ as a function of $x$.

The Absolute Value Function

• Definition: The absolute value or magnitude of a real number $x$ is defined piecewise as:     $|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$

• Examples of Evaluation:     * $|5| = 5$     * $|-\frac{4}{7}| = \frac{4}{7}$     * $|0| = 0$

• Properties 0.1.4: For real numbers $a$ and $b$:     * (a) $|-a| = |a|$ (A number and its negative have the same absolute value).     * (b) $|ab| = |a||b|$ (Absolute value of a product is the product of absolute values).     * (c) $|\frac{a}{b}| = \frac{|a|}{|b|}$, for$b \neq 0 (Absolute value of a ratio is the ratio of absolute values).\n    * (d) |a + b| \le |a| + |b| (The triangle inequality).\n* **Geometric Representation:** The graph of f(x) = |x|$is V-shaped, formed by combining the ray$y = -x$(for$x < 0$) and the ray$y = x$(for$x \ge 0).\n* **Square Root Relationship:** The symbol \sqrt{x}$always denotes the **positive** square root of$x. There is a critical identity relating square roots and absolute values:\n    \sqrt{x^2} = |x|\n* **Warning on Square Roots:** To denote a negative square root, one must write -\sqrt{x}$. For example,$\sqrt{9} = 3$. It is an error to state$\sqrt{9} = \pm 3.\n\n## Piecewise-Defined Functions\n\n* **Definition:** A function whose formula changes depending on the value of the input x.\n* **Breakpoints:** The values of x where the formula used to calculate the function changes.\n* **Example 4:** A function defined by:\n    f(x) = \begin{cases} 0, & x \le -1 \ \sqrt{1 - x^2}, & -1 < x < 1 \ x, & x \ge 1 \end{cases}\n    * The graph consists of a horizontal ray y = 0$on$(-\infty, -1]$, a semicircle$y = \sqrt{1 - x^2}$on$(-1, 1)$, and a ray$y = x$on$[1, +\infty).\n    * **Graphing Conventions:** Use solid dots to indicate points included in the graph at breakpoints (e.g., at x = 1, f(1) = 1$) and open circles to indicate points excluded (e.g., the end of the semicircle at$x = 1).\n* **Example 5 (Wind Chill Index):** The wind speed v$in$\text{mi/h}$affects the perceived temperature$W$. At an air temperature of$32^\circ \text{F}, the empirical formula for the wind chill index is:\n    W = \begin{cases} 32, & 0 \le v \le 3 \ 55.628 - 22.07v^{0.16}, & 3 < v \end{cases}\n\n## Domain and Range\n\n* **Definitions:**\n    * **Domain:** The set of all allowable inputs (x-values).\n    * **Range:** The set of all resulting outputs (y$-values) as$x varies over the domain.\n* **Natural Domain (Definition 0.1.5):** If a function is defined by a formula and no domain is explicitly stated, the domain is understood to be all real numbers for which the formula yields a real value.\n* **Visual Interpretation:** The domain can be found by projecting the graph of y = f(x)$onto the$x$-axis; the range is found by projecting the graph onto the$y-axis.\n* **Example 6 (Natural Domain Calculations):**\n    * (a) f(x) = x^3$: Domain is$(-\infty, +\infty).\n    * (b) f(x) = \frac{1}{(x-1)(x-3)}$: Domain is$(-\infty, 1) \cup (1, 3) \cup (3, +\infty)$, excluding$x = 1$and$x = 3 where division by zero occurs.\n    * (c) f(x) = \tan(x) = \frac{\sin(x)}{\cos(x)}$: Domain excludes values where$\cos(x) = 0$, which are odd integer multiples of$\frac{\pi}{2}.\n    * (d) f(x) = \sqrt{x^2 - 5x + 6}$: Requires$x^2 - 5x + 6 \ge 0$. Factoring gives$(x-3)(x-2) \ge 0$, which is satisfied for$x \le 2$or$x \ge 3$. Domain is$(-\infty, 2] \cup [3, +\infty).\n* **Algebraic Simplification and Domain:** Simplifying a function by canceling factors can change its domain.\n    * **Example 7:** The function f(x) = \frac{x^2 - 4}{x - 2}$is undefined at$x = 2$. Although algebraic simplification results in$f(x) = x + 2$, the domain must still exclude$x = 2$. Geometrically, the graph has a "hole" at the point$(2, 4).\n* **Range Determination via Algebra:** \n    * **Example 8(b):** For f(x) = \frac{x+1}{x-1}$, to find the range, solve for$x$in terms of$y:\n        y = \frac{x+1}{x-1} \implies xy - y = x + 1 \implies x(y - 1) = y + 1 \implies x = \frac{y+1}{y-1}\n        This shows that y = 1$is excluded from the range because it would cause division by zero. Range is$(-\infty, 1) \cup (1, +\infty).\n\n## Applied Problems: Domain and Range\n\n* **Example 9 (Box Construction):** An open box is made by cutting squares of side x$from a$16\, \text{in} \times 30\, \text{in} cardboard sheet.\n    * (a) Volume formula: V(x) = (16 - 2x)(30 - 2x)x = 480x - 92x^2 + 4x^3.\n    * (b) Domain: Since lengths must be positive and the shortest side is 16\, \text{in}$,$2x$cannot exceed$16$. Thus, the domain is$0 \le x \le 8.\n    * (c) Range: Based on the graph, the volume V$is approximately in the range$[0, 725]\, \text{in}^3.\n    * (d) Maximum Volume: Occurs for an x$value between$3\, \text{in}$and$4\, \text{in}$, yielding a maximum volume of approximately$725\, \text{in}^3.\n* **Example 10 (Radar Tracking):** A car travels at a constant speed of 100\, \text{ft/s} between 8:05 A.M. and 8:06 A.M.\n    * Defining t = 0$as 8:05 A.M. and$t = 60 as 8:06 A.M., the distance traveled is given by:\n        D(t) = 100t, \, 0 \le t \le 60 (in seconds).\n\n## Issues of Scale and Units\n\n* **Preserving Shape:** In geometry, equal units of length must be used on both axes to present the true shape (e.g., to prevent a circle from appearing as a squashed ellipse).\n* **Compression for Clarity:** When the range of a function is much larger than its domain (e.g., y = x^2$over$[-10, 10]$where$y$goes up to$100$), compressing the$y-axis scale is often necessary to fit the graph on a page.\n\n## Exercises for Practice\n\n* **Problem 7:** Find specific values for piecewise functions:\n    * (a) f(x) = 3x^2 - 2$for all$x.\n    * (b) f(x) = \begin{cases} \frac{1}{x}, & x > 3 \ 2x, & x \le 3 \end{cases}\n* **Problem 8:** Find values and composite inputs for piecewise functions:\n    * (a) g(x) = \frac{x+1}{x-1}.\n    * (b) g(x) = \begin{cases} \frac{1}{x}, & x \ge 1 \ 3, & x < 1 \end{cases}$$