General Mathematics Midterm Review: Functions, Rational Functions, Piecewise-Defined Functions, and Operations (Vocabulary)
Functions and Their Graphs
Functions arise from a relation between two sets of data where one set depends on the other. They describe relationships, trends, and predictions (e.g., mailing cost depends on weight; velocity depends on force).
Velocity is the rate of change of position with respect to time; mathematics of velocity uses functions and graphs.
Key terms:
Correspondence
Relation: a set of ordered pairs
Function: a relation that assigns to each element of a domain X exactly one element in codomain Y
Independent variable: the input (often x)
Dependent variable: the output (often y or f(x))
Vertical Line Test: a graph represents a function if no vertical line intersects the graph more than once
Notation: conventional function notation is f(x)=y where
f is the function
x is the independent variable (input)
y is the dependent variable (output)
f(x) is read as "f of x" and is the rule applied to input x to give output y
Important distinction: not all relations are functions; all functions are relations.
Example mappings in sets (from the transcript):
Guard set G and Bigs set B with a set of pairings R = {(Max, Jong), (Max, Dave), (Nathan, Jong), (Nathan, Dave), (Alex, Dave), (Alex, Karlo)}
Renaming/relabelling can yield M = {(1,4), (1,5), (2,4), (2,5), (3,5), (3,6)} and T = {(1,6), (2,6), (3,4)}
Observation: R and M are relations; T is a relation where every element of X is paired with exactly one element of Y, illustrating a special type of relation: a function.
Functions as models: functions model real-world processes and can approximate behavior of variables.
In business/economics, functions model income, costs, etc., to describe relationships and predict outcomes.
Definition of a Function
A function f is a relation that assigns each element of a set X to exactly one element of another set Y.
A function may be one-to-one or many-to-one; it cannot be one-to-many.
Notation and interpretation:
For every input x, there is exactly one output y, given by f(x).
Graphically, the function is the rule that maps x to f(x).
Example: A function can be described by an equation like f(x)=2x+5
This can be written as y=2x+5; y and f(x) are outputs after applying the rule to x.
A simple input/output model: INPUT: (x) → RULE: "double x then add 5" → OUTPUT: f(x)
Example values: for x=2, f(2)=2(2)+5=9; for x=-4, f(-4)=2(-4)+5=-3.
In tabular form, a function maps each x to a unique f(x):
If f(x) = 2x+5, then a table with x-values (2,0,1,7,…) yields corresponding f(x) values.
Example of non-unique x mapping: the graph of y = x^2 yields f(-1)=1 and f(1)=1, showing multiple x-values can map to the same y-value and still be a function (the key is that each x maps to exactly one y).
1.3 Distinguishing Functions from Relations
Tests to tell whether a relation is a function:
1) Given coordinates (table of values): check that no two ordered pairs share the same x-coordinate. If any x repeats with different y, it is not a function.
2) Given a graph: use the Vertical Line Test. If any vertical line intersects the graph more than once, the relation is not a function.Examples from the transcript:
P = {(0,1), (2,1), (3,3), (4,7), (5,9)}: x-values are all unique; P is a function.
N = {(0,-1), (4,4), (1,6), (0,2), (2,8)}: x=0 occurs with two y-values (-1 and 2); not a function.
A table with distinct x-values across all rows is a function, regardless of repeated y-values.
Graph-based check: a graph is a function if there is no vertical line that intersects it more than once.
1.4 Functions as Models
Functions are used to model real-life situations rather than to produce exact forecasts in every case.
Examples of real-world modeling:
Bayanihan.org membership data (years 2013–2017) modeled to understand trends in volunteer numbers.
First modeling attempt: linear model f(x) = 1.5x fits the first two years well but fails for later years.
Improved modeling: a quadratic model f(x) = 0.5x^2 + 1 fits all five data points better and can be used for prediction.
Example prediction: If x represents a year index corresponding to 2019, and x = 7, then f(7) = 0.5*(7)^2 + 1 = 25.5, implying 25,500 volunteers (in thousands as labeled in the example).
Key ideas about piecewise behavior and nonlinearity in real data:
In many real situations, a single simple function is not adequate; piecewise definitions or higher-order models may be necessary.
Absolute value functions and piecewise definitions often arise when modeling thresholds or regime changes.
2 Evaluating Functions
Evaluating a function means plugging an argument (input value) into the function to compute the output.
The argument is the input value x; the function value is f(x).
Example: For g(x) = 4x − 3, evaluate at various x-values:
g(2) = 4(2) − 3 = 5
g(-4) = 4(-4) − 3 = −19
g(-1) = 4(-1) − 3 = −7
g(0) = 4(0) − 3 = −3
Example: For p(x) = 2x^2, evaluate at x ∈ {1, -2, 0, -3}:
p(1) = 2(1)^2 = 2
p(-2) = 2(−2)^2 = 8
p(0) = 0
p(-3) = 2(−3)^2 = 18
Domain of a function: the set of all possible x-values (arguments) for which the function is defined and yields a real output.
Important notes:
Some functions have all real numbers as permissible inputs (domain = \mathbb{R}), e.g., f(x)=2x+7.
Some functions restrict inputs due to division by zero or square roots of negatives (e.g., g(x)= rac{1}{x} has domain ext{dom}(g)=\mathbb{R}ackslash\{0\}).
For some simple fractions, e.g., b(x)= rac{x+2}{9}, the domain is all real numbers except where the denominator is zero (in this case the denominator is never zero, so dom is ext{dom}(b)= ext{R}).
Examples from the transcript:
Domain of f(x)=2x+7: ext{dom}(f)=\mathbb{R}
Domain of g(x)= rac{1}{5}: denominator never zero, so ext{dom}(g)= ext{R}
Domain of b(x)= rac{x+2}{9}: exclude value that makes denominator zero: here none, but for general case, solve $x+2=0$ if applicable; if you had b(x)= rac{1}{x+2}, then ext{dom}= \, ext{R}ackslash{-2\, ext{}}
Special case: nested radicals such as h(x)=\sqrt{\sqrt{x}+5} require
\sqrt{x}+5\ge 0\Rightarrow x\ge -5, giving domain [-5,\infty).
Piecewise-Defined Functions
A piecewise-defined function is described by multiple sub-functions, each applying on a specified interval (part of the domain).
Example: Miss Lea’s salary S as a function of sales x
S1(x) = 20{,}000 + 0.04x, for 0 ≤ x ≤ 50{,}000
S2(x) = 20{,}000 + 0.10x, for x > 50{,}000
This creates a piecewise function S(x) with a break (threshold) at x = 50{,}000. The domain is restricted to nonnegative x (x ≥ 0).
Graphing a piecewise function:
Each sub-function is drawn on its respective sub-domain.
Endpoints at sub-domain boundaries are indicated by solid or hollow circles depending on whether the boundary value is included in the domain for that sub-function.
Example 1: Piecewise linear sub-functions
For a graph defined by k(x) = { x − 1, x < −1; 7, −1 ≤ x < 2; 6 − x, x ≥ 2 }
Graph each piece on its domain; left piece ends with a hollow circle at x = −1, middle piece has a solid circle at x = −1 and a hollow circle at x = 2, right piece has a solid circle at x = 2.
Example 2: A more complex piecewise graph with four sub-parts over different intervals.
Applications of piecewise-defined functions:
Bike rental: cost F is charged per hour or fraction thereof, up to 4 hours. Sub-functions: F1 = 100 for 0 < x ≤ 1, F2 = 200 for 1 < x ≤ 2, F3 = 300 for 2 < x ≤ 3, F4 = 400 for 3 < x ≤ 4.
Batario karaoke room: cost T includes a flat rate of 500 plus per-hour rates that change after certain thresholds:
T1(x) = 500x + 500 for 0 < x ≤ 2
T2(x) = 300(x − 2) + 1500 for 2 < x ≤ 4 (equivalently, 300(x − 2) + 1500)
T3(x) = 200(x − 4) + 2100 for x > 4
Piecewise definition summary for the Batarios:
T(x) = \begin{cases} 500x + 500, & 0 < x \le 2 \[6pt] 300(x-2) + 1500, & 2 < x \le 4 \[6pt] 200(x-4) + 2100, & x > 4 \end{cases}
Another important point: piecewise-defined functions are absolute-value-like and can model sudden changes or thresholds in real-world processes.
3 Operations on Functions
Addition: For functions f and g with domains A and B, the sum (f+g)(x) = f(x) + g(x) with domain A ∩ B.
Example results from the transcript:
If f(x) = 5x + 2 and g(x) = 3 − 2x, then (f+g)(x) = 3x + 5.
Subtraction: (f − g)(x) = f(x) − g(x) with the domain A ∩ B.
Example: If f(x) = 2x − 3 and g(x) = 5 − 4x, then (f − g)(x) = 6x − 8.
Multiplication: (f g)(x) = f(x) g(x) with domain A ∩ B.
Example: If f(x) = 3x + 1, g(x) = x − 3, h(x) = 2x − 5, then (f g)(x) = (3x+1)(x−3) = 3x^2 − 8x − 3.
Division: (f/g)(x) = f(x)/g(x) with domain restricted to x ∈ A ∩ B and g(x) ≠ 0.
Composition: (f ∘ g)(x) = f(g(x)) with domain constraints: g(x) must lie in the domain of f.
Example: If f(x) = 2x − 3 and g(x) = x^2,
(f ∘ g)(x) = f(g(x)) = 2(x^2) − 3 = 2x^2 − 3
(g ∘ f)(x) = g(f(x)) = g(2x − 3) = (2x − 3)^2
Composition is not commutative: (f ∘ g) ≠ (g ∘ f) in general.
Chapter II: Rational Functions
Rational functions are quotients of polynomial functions with a denominator that is not identically zero and cannot be zero at the chosen input: if
F(x) = \frac{P(x)}{Q(x)}, \quad Q(x) \neq 0,
then F is a rational function.Rational expressions (algebraic fractions) are quotients of polynomials: \frac{P(x)}{Q(x)} with P(x) and Q(x) polynomials; we must exclude inputs that make Q(x) = 0.
Real-world modeling: rational functions model quantities that involve division of polynomials (e.g., pricing, rates, times, mixtures).
Important ideas about number sense and domain:
The domain excludes values that make the denominator zero; these excluded values are potential extraneous inputs when solving equations.
The domain is typically a subset of \mathbb{R} (or sometimes all real numbers except a finite set).
Example intuition: For f(x) = \frac{1}{x}, the domain is \mathbb{R} \setminus {0}; inputs approaching 0 lead to arbitrarily large outputs (vertical asymptote at x=0).
Travel-time example (rational function in real-world):
Suppose distance d is fixed and speed s varies, total time T(s) = travel time + fixed time = ( \frac{d}{s} + 4 ).
This is a rational-like model because of the division by s; the graph has a vertical asymptote at s = 0 and decreases as s increases, approaching the horizontal asymptote T = 4 as s grows large.
Another example: ticket pricing from a venue:
If total cost is the venue charge divided among attendees plus a fixed snack cost, the price per ticket is g(x) = \frac{3000}{x} + 50, where x is the number of attendees.
This illustrates a rational function where the denominator (x) must be nonzero (x ≠ 0) and the function behavior as x changes.
Rational Functions in the Real World
Rational functions model scenarios where quantities depend on a ratio of polynomials (e.g., rates, prices per unit, total costs per number of items).
The concept of extraneous roots arises when solving rational equations: solutions must also lie in the domain (i.e., must not make any denominator zero).
Rational Functions: Solving Rational Equations and Inequalities
Rational equations involve solving for x in equations that include rational expressions; the steps often include: factoring, finding a common denominator (LCD), and checking extraneous roots.
Extraneous roots: values that satisfy the manipulated equation but are not in the domain (because they make a denominator zero).
Factoring often helps cancel common factors, but you must ensure cancellations do not remove domain restrictions.
Example solving rational equations (illustrative, not fully verbatim from transcript):
If you have an equation like ( \frac{x^2+6x+9}{x+3} = 5 ), factor the numerator as ( (x+3)^2 ) and cancel to obtain a simpler equation; solve, then check domain to avoid extraneous roots.
Another common pattern: ( \frac{2x^2-5x-12}{x^2+7x-8} = \frac{x-4}{x-1} ). After factoring, cancel common factors only where valid and check domain restrictions (x ≠ 1, -8 in this example).
Rational inequalities: solve by turning to a single rational expression, identifying critical values (zeros of numerator and denominator), and testing sign intervals.
Step-by-step method (as given in transcript):
1) Move everything to one side; 2) express numerator and denominator in factored form; 3) identify critical values; 4) partition the real line into intervals; 5) determine the sign of each factor in each interval; 6) determine which intervals satisfy the inequality; 7) assemble the solution in interval notation.Example result (from transcript): For the inequality
\frac{x^2 - x - 12}{x^2 + 8x + 7} > 0,
the solution set is (-\infty, -7) \cup (-3, -1) \cup (4, \infty).Another example with ≤0 yields a solution like (-5,-2] \cup (3,5].
Graphing Rational Functions
Tables of values: an important method to prepare to graph rational functions without a graphing calculator.
Intercepts: x-intercepts (where output y = 0) and y-intercepts (where x = 0).
Vertical asymptotes: occur where the denominator is zero (domain restrictions). They are lines x = a where the function is undefined.
Horizontal vs Slant asymptotes:
If deg(numerator) < deg(denominator), horizontal asymptote is y = 0.
If deg(numerator) = deg(denominator), horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).
If deg(numerator) > deg(denominator), no horizontal asymptote; there is a slant (oblique) asymptote given by the polynomial part of the quotient when the division is performed.
Steps to graph a rational function:
1) Find vertical asymptotes by solving denominator = 0.
2) Find horizontal or slant asymptote as described above.
3) Find x- and y-intercepts by setting y = 0 and x = 0 respectively.
4) Plot several points on each interval separated by the asymptotes.
5) Sketch the curve, keeping in mind asymptotes guide the shape of the graph.Examples (from transcript):
f(x) = 1/(x) has vertical asymptote at x = 0 and horizontal asymptote y = 0; the graph approaches the axes but never touches the asymptotes.
f(x) = (2x^2 − 2)/(x^2 − 9) has vertical asymptotes at x = −3 and x = 3; horizontal asymptote y = 2 (since degrees are equal and leading coefficients yield 2/1 = 2).
f(x) = (x^2 − 2x − 8)/(x+1)(x−3) has a vertical asymptote at x = −1 and a slant asymptote y = x − 3 (since degree numerator > degree denominator, performing long division yields a linear quotient for the asymptote).
Applications of Rational Functions
The envelope of rational functions in real-world problems includes time, cost, velocity, and other rates where outputs depend on ratios.
Example: Travel time vs. speed (with fixed distance d and a fixed stop time t0):
Total time T(s) = ( \frac{d}{s} + t_0 ) where s is speed.
For d = 200 km and t0 = 4 h, T(s) = ( \frac{200}{s} + 4 ).
Example: Finances and tickets (as function of attendees x):
Total cost or price per ticket is a rational function depending on the number of attendees.
Example: If venue rental is 3000 and snacks cost 50 per attendee, price per attendee g(x) = ( \frac{3000}{x} + 50 ).
Graphical interpretation emphasizes that rational functions can exhibit vertical asymptotes (domain gaps) and horizontal/slant asymptotes reflecting long-run behavior.
Check Your Understanding (Key Concepts to Remember)
A rational function is the quotient of two polynomials with a nonzero denominator: F(x) = \frac{P(x)}{Q(x)}, \quad Q(x) \neq 0.
The domain of a rational function excludes inputs x that make the denominator zero.
When solving rational equations, always check for extraneous roots by testing those roots in the original equation and ensuring they do not violate the domain.
When solving rational inequalities, partition the real line using critical values (zeros of numerator and denominator) and test the sign of the rational expression in each interval.
Graphing rational functions requires identifying vertical, horizontal, and slant asymptotes as guides to the shape of the graph.
Summary (Chapter Summary from Transcript)
A rational function is the quotient of polynomial functions with a denominator that may not be zero: F(x)=\frac{P(x)}{Q(x)}, \quad Q(x) \neq 0.
The domain of a rational function is the set of x for which the denominator is not zero.
Rational functions differ from rational equations and rational inequalities because the former are functions (mapping from x to y), while the latter are equations/inequalities involving rational expressions.