Functions and Graphs

Objectives and Textbook Overview

• Reference Text: Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences (Thirteenth Edition) by Haeussler, Paul, and Wood.

• Chapter Subject: Functions and Graphs.

• Chapter Objectives:

  • To understand what functions and domains are.

  • To introduce different types of functions.

  • To become familiar with the shapes of the graphs of six basic functions.

Fundamentals of Functions, Domain, and Range

• Definition of a Function: A function assigns each input number to exactly one output number.

• Definition of Domain: The set of all input numbers to which the function applies is called the domain of the function.

• Definition of Range: The set of all output numbers produced by evaluating the function across its domain is called the range.

Methods for Finding the Domain of Algebraic Functions

• General Rule for Finding Domain:

  • To find the domain of a function, first identify the excluded values (values for which the expression is undefined, such as division by zero).

  • The domain is the set of real numbers $\mathbb{R}$ from which all excluded values are removed.

• Example 1: Domain and Evaluation of a Polynomial Function

  • Given function:     $g(x) = 3x^2 - x + 5$

  • Finding Domain: Any real number can be used for $x$, so the domain of $g$ is all real numbers:     $D(g) = (-\infty, +\infty)$

  • Evaluation Exercises:

  • Find $g(z)$       $g(z) = 3z^2 - z + 5$

  • Find $g(r^2)$       $g(r^2) = 3(r^2)^2 - r^2 + 5$       $g(r^2) = 3r^4 - r^2 + 5$

  • Find $g(x + h)$       $g(x + h) = 3(x + h)^2 - (x + h) + 5$       $g(x + h) = 3(x^2 + 2xh + h^2) - x - h + 5$       $g(x + h) = 3x^2 + 6xh + 3h^2 - x - h + 5$

• Example 2: Domain of a Rational Function

  • Given function:     $f(x) = \frac{2x}{3x - 4}$

  • Finding Excluded Values: Set the denominator equal to zero:     $3x - 4 = 0$     $3x = 4$     $x = \frac{4}{3}$

  • Domain Solution:     $D(f) = \mathbb{R} \setminus \left\{\frac{4}{3}\right\} = \left(-\infty, \frac{4}{3}\right) \cup \left(\frac{4}{3}, +\infty\right)$

Determining Domain and Range from Graphs

• Procedure for Graphical Identification:

  • To find the domain of a function from its graph, look for all inputs on the $x$-axis that correspond to points on the graph.

  • To find the range of a function from its graph, look for all outputs on the $y$-axis that correspond to points on the graph.

• Example 1: Continuous Curve with Closed Endpoints   !Graph of y = f(x) showing domain [-2, 1] and range [0, 3]

  • Graph features: Starts at solid endpoint $(-2, 0)$, curves up through $(0, 2)$, and ends at solid endpoint $(1, 3)$.

  • Domain:     $\{x \mid -2 \le x \le 1\}$     $[-2, 1]$

  • Range:     $\{y \mid 0 \le y \le 3\}$     $[0, 3]$

• Example 2: Line Segment with Open and Closed Endpoints   ![Graph of y = f(x) showing domain (-2, 1] and range -1, 2)

  • Graph features: Open endpoint at $(-2, 2)$, passing through $(0, 0)$, ending at solid endpoint $(1, -1)$.

  • Domain:     $\{x \mid -2 < x \le 1\}$     $(-2, 1]$

  • Range:     $\{y \mid -1 \le y < 2\}$     $[-1, 2)$

Special Types of Functions

• Constant Function:

  • Definition: A function of the form $h(x) = c$, where $c$ is a constant real number, is a constant function.

  • Domain: The domain of $h$ is all real numbers $(-\infty, +\infty)$.

  • Example: Let $h(x) = 2$     $h(10) = 2$     $h(-387) = 2$     $h(x + 3) = 2$

• Rational Functions:

  • Definition: A function $f(x)$ is a rational function if both the numerator and denominator are polynomial functions.

  • Rational Example A:     $f(x) = \frac{x^2 - 6x}{x + 5}$     This is rational because both $x^2 - 6x$ and $x + 5$ are polynomials.

  • Rational Example B:     $g(x) = 2x + 3$     This is rational because it can be written as a fraction over $1$:     $2x + 3 = \frac{2x + 3}{1}$

• Absolute-Value Function:

  • Definition: The absolute-value function $f(x) = |x|$ is defined piece-wise as:     $|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$

Inverse Functions

• Definition of Inverse Functions:

  • Let $f$ and $g$ be two functions such that:     $f(g(x)) = x \quad \text{for every } x \text{ in the domain of } g$     $g(f(x)) = x \quad \text{for every } x \text{ in the domain of } f$

  • The function $g$ is the inverse of the function $f$ and is denoted by $f^{-1}$ (read "$f$-inverse").

  • Identity equations:     $f(f^{-1}(x)) = x$     $f^{-1}(f(x)) = x$

  • Domain-Range Relationship: The domain of $f$ is equal to the range of $f^{-1}$, and the range of $f$ is equal to the domain of $f^{-1}$.

• Example: Verifying Inverse Functions

  • Show that $f$ and $g$ are inverse functions:     $f(x) = 4x - 7$     $g(x) = \frac{x + 7}{4}$

  • Evaluation of $f(g(x))$:     $f(g(x)) = f\left(\frac{x + 7}{4}\right)$     $f(g(x)) = 4\left(\frac{x + 7}{4}\right) - 7$     $f(g(x)) = x + 7 - 7$     $f(g(x)) = x$

  • Evaluation of $g(f(x))$:     $g(f(x)) = g(4x - 7)$     $g(f(x)) = \frac{(4x - 7) + 7}{4}$     $g(f(x)) = \frac{4x}{4}$     $g(f(x)) = x$

  • Conclusion: Since $f(g(x)) = g(f(x)) = x$, $f$ and $g$ are inverse functions.

Graphs and Intercepts in Rectangular Coordinates

• Rectangular Coordinate System: Provides a geometric framework to graph equations in two variables.

• Definition and Rules of Intercepts:

  • To find $x$-intercepts, look for points where the graph crosses or intersects the $x$-axis (set $y = 0$).

  • To find $y$-intercepts, look for points where the graph crosses or intersects the $y$-axis (set $x = 0$).

  • Intercept limit: A function can have more than one $x$-intercept, but at most one $y$-intercept.

• Example 1: Quadratic Function Intercepts

  

  • Equation: $y = x^2 + 2x - 3$

  • $x$-intercepts: Points where curve crosses $x$-axis.

  • $y$-intercept: Point where curve crosses $y$-axis.

• Example 2: Reading Intercepts from Graph

  • For graph of $f(x)$, the $x$-intercepts are $(-3, 0)$, $(-1, 0)$, and $(2, 0)$.

  • The $y$-intercept is $(0, -6)$.

• Example 3: Intercepts and Graph of a Linear Function

  

  • Given equation: $y = 2x + 3$

  • Finding $x$-intercept (set $y = 0$):     $0 = 2x + 3$     $2x = -3$     $x = -\frac{3}{2}$     $x\text{-intercept: } \left(-\frac{3}{2}, 0\right)$

  • Finding $y$-intercept (set $x = 0$):     $y = 2(0) + 3$     $y = 3$     $y\text{-intercept: } (0, 3)$

  • Table of Values for $y = 2x + 3$:

  • $x = 0 \implies y = 3$

  • $x = -\frac{3}{2} \implies y = 0$

  • $x = \frac{1}{2} \implies y = 4$

  • $x = -\frac{1}{2} \implies y = 2$

  • $x = 1 \implies y = 5$

  • $x = -1 \implies y = 1$

  • $x = 2 \implies y = 7$

  • $x = -2 \implies y = -1$

• Example 4: Vertical Line Graph

  

  • Equation: $x = 3$

  • $x$-intercept: $(3, 0)$

  • $y$-intercept: There is no $y$-intercept because $x$ cannot be $0$

  • Table of Values:

  • $(3, 0)$

  • $(3, 3)$

  • $(3, -2)$

Algebraic and Graphical Symmetry: Even and Odd Functions

• Algebraic Definitions:

  • Even Function: Function $f$ is even if $f(-x) = f(x)$ for all $x$ in the domain of $f$.

  • Odd Function: Function $f$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$.

• Example: Determining Even/Odd Symmetry

  • Given function: $h(x) = x^5 + 1$

  • Testing Evenness:     $h(-x) = (-x)^5 + 1$     $h(-x) = -x^5 + 1$     Since $h(-x) \neq h(x)$, the function is not even.

  • Testing Oddness:     $-h(x) = -(x^5 + 1)$     $-h(x) = -x^5 - 1$     Since $h(-x) \neq -h(x)$, the function is not odd.

  • Conclusion: The function $h(x) = x^5 + 1$ is neither even nor odd.

• Graphical Symmetry Principles:

  • The graph of an even function ($f(-x) = f(x)$) is symmetric with respect to the $y$-axis.

  • The graph of an odd function ($f(-x) = -f(x)$) is symmetric with respect to the origin.