General Mathematics – Consolidated Study Notes

FUNCTIONS AND THEIR GRAPHS

Objectives

• Define a function, domain, and range.

• Use the vertical line test to classify relations.

• Evaluate functions for given inputs.

• Graph and interpret linear & quadratic functions.

• Relate graphs to real-world problems.

Definition of a Function

• A function $f$ is a rule assigning exactly one output $f(x)$ to each input $x$ in its domain $A$:
  $f:A\rightarrow B$

• Domain = set of permissible $x$–values.

• Range = set of attainable $f(x)$–values.

Core Families of Algebraic Functions

• Polynomial: $f(x)=a<em>nx^n+\dots+a</em>1x+a_0$, $n\in\mathbb N$.

• Rational: $f(x)=\dfrac{p(x)}{q(x)},\;q(x)\neq0$.

• Root / Radical: $f(x)=\sqrt[n]{p(x)}$.

• Implicit algebraic: defined by an equation in $x$ & $y$ (e.g. $x^2+y^2=9$).

Properties (illustrative)

• Domain restrictions arise where square-roots, denominators, or fractional exponents become invalid.
  • Example: $f(x)=\sqrt{x} \Rightarrow x\ge0$.

• Behaviours
  • Polynomials: smooth, continuous.
  • Rationals: discontinuities, vertical/horizontal asymptotes.

Vertical Line Test (VLT)

• "A graph represents a function iff no vertical line intersects it more than once."

• Examples

  • Parabola $y=x^2$ ✓

  • Circle $x^2+y^2=9$ ✗ (fails VLT)

  • Line $y=2x-3$ ✓

  • Sine curve on one period ✓ (unique $y$ for each $x$)

Evaluating Functions – Algorithm

1. Identify the expression $f(x)$.

2. Substitute the given input.

3. Simplify using $\text{PEMDAS/BODMAS}$.

Worked Examples

• Linear: $f(x)=3x+5;\;f(4)=3(4)+5=17$.

• Quadratic: $g(x)=x^2-4x+3;\;g(-2)=(-2)^2-4(-2)+3=15$.

• Fractional: $h(x)=\dfrac{x+1}{x-2};\;h(3)=\dfrac{4}{1}=4$.
  • Undefined at $x=2$ (division by $0$).

Common Pitfalls

• Dropping parentheses with negatives: $f(-3)=2(-3)^2\neq2\cdot-3^2$.

• Forgetting to test for zero denominators.

Graphing Basics

• Steps

  1. Identify function type & key traits (intercepts, asymptotes, vertex…).

  2. Choose representative $x$–values (include negatives & zero).

  3. Compute $y$–values.

  4. Plot $(x,y)$ pairs.

  5. Connect smoothly, respecting known behaviour.

• Key features to mark

  • $x$– & $y$–intercepts

  • Slope (linear)

  • Vertex (quadratic)

  • Asymptotes (rational, exponential, log)

  • Symmetry (even/odd/periodic)

Linear Example

$y=2x+1$ — use $x=-2,-1,0,1,2$; plot, draw straight line.

Quadratic Example

$f(x)=x^2-4x+3$

• Table: $x=0,1,2,3,4 \Rightarrow f(x)=3,0,-1,0,3$.

• Vertex at $x=2$ (minimum $-1$).

Real-World Application (T-Shirt Company)

• Cost function: $C(x)=0.20x^2+5x+200$ (values inferred from text).

• Revenue: $R(x)=12x$.

• Profit: $P(x)=R(x)-C(x)=12x-(0.20x^2+5x+200)= -0.20x^2+7x-200$.
  • Profit at $x=50$ shirts: $P(50)= -0.20(2500)+350-200= -500+150= -350$ (loss).

• Break-even: solve $P(x)=0\;\Rightarrow\;x\approx28.9$ → need $29$ shirts.

RATIONAL EQUATIONS, FUNCTIONS & INEQUALITIES

Objectives

• Solve rational equations / inequalities.

• Graph rational functions & locate asymptotes.

• State domain & range.

• Apply concepts to pipes, motion, business, etc.

Prerequisite Skills

• Fraction operations & LCD.

• Factoring (common, quadratic, difference of squares).

• Solving linear & quadratic equations.

• Domain restrictions (denominator $\neq0$).

• Cross-multiplication; checking extraneous roots.

Rational Equations – Prototypes & Methods

1. Simple form: $\dfrac{3}{x}=4$ → LCD $x$ → $3=4x\Rightarrow x=\dfrac34$.

2. Two rational terms: $\dfrac{x}{x+2}=\dfrac34$

   • LCD $(x+2)\cdot4$.

   • Cross-multiply → quadratic; factor & test restrictions.

3. Quadratic rational: $\dfrac{x+1}{x-2}=x$ → cross-multiply $x+1=x(x-2)$.

4. Fractions both sides: $\dfrac{x}{x-1}+\dfrac{2}{x-1}=5$ → combine numerators.

5. Application (pipes): $\dfrac1{4}+\Bigl(-\dfrac1{6}\Bigr)=\dfrac1t$.

Worked Example (Intermediate)

$\frac1x+\frac1{x+2}=\frac34$

• LCD $4x(x+2)$ → $4(x+2)+4x=3x(x+2)$.

• Expand $4x+8+4x=3x^2+6x$ → $3x^2-2x-8=0$ → factor $(3x+4)(x-2)=0$.

• Roots $x=-\tfrac43$ (allowed), $x=2$ (reject; $x+2=0$?).

Rational Functions – Anatomy

• General form $f(x)=\dfrac{p(x)}{q(x)},\;q(x)\neq0$.

• Domain: all real $x$ where $q(x)\neq0$.

Graph Features

• Vertical asymptotes: zeros of $q(x)$ (after canceling holes).

• Horizontal asymptote: compare degrees of $p$ & $q$.

• Holes: common factors canceled.

Sample Catalog

1. $f(x)=\dfrac1x$: VA $x=0$, HA $y=0$, odd symmetry.

2. $g(x)=\dfrac{2x+1}{x-3}$: VA $x=3$, oblique asymptote $y=2$ (same degree).

3. $h(x)=\dfrac{x^2-1}{x-1}=x+1$ with hole at $x=1$.

Rational Inequalities – Strategy

1. Rewrite to zero on one side.

2. Factor numerator & denominator.

3. Mark critical points (zeros & undefineds) on number line.

4. Test intervals; respect <,\le,>,\ge.

5. Express solution set, exclude undefined points.

Example

Solve $\dfrac{x+2}{x-1}\le0$

• Critical points $x=-2,1$ (1 undefined).

• Test intervals:
  • $(-\infty,-2]\Rightarrow$ numerator <0, denominator <0 → quotient >0 ✗
  • $(-2,1)$ → $+/-$ → <0 ✓ • $(1,\infty)$ → $+/+$ → >0 ✗

• Include $x=-2$ (makes zero). Exclude $x=1$. Solution $(-2,1)$.

Mixed Applied Problems

• Work: Amy (6 h) & Ben (4 h). $\frac1x=\frac16+\frac14\Rightarrow x=\frac{12}{5}=2.4\text{ h}$.

• Motion: 60 km downstream (3 h), upstream (5 h). Solve system \begin{cases}b+c=20\b-c=12\end{cases} → $b=16\,\text{km/h},\;c=4\,\text{km/h}$.

EXPONENTIAL & LOGARITHMIC FUNCTIONS

Objectives

• Describe properties of $a^x$ & $\log_b(x)$.

• Solve related equations.

• Graph and identify asymptotes/intercepts.

• Apply models of growth, decay, pH, finance.

Essential Prior Knowledge

• Algebraic manipulation & order of operations.

• Exponent laws: $a^m a^n=a^{m+n}$, $\bigl(a^m\bigr)^n=a^{mn}$, $a^{-n}=\dfrac1{a^n}$.

• Function notation, inverses, graph transformations.

• Scientific notation.

Exponential Functions

• General form $f(x)=ab^{x}$ with a\neq0,\;b>0,\;b\neq1.

• Growth when b>1; decay when 0<b<1.

• Domain $(-\infty,\infty)$; range $(0,\infty)$.

• Horizontal asymptote $y=0$ (unless shifted).

• Key point $(0,a)$.

Sample Evaluation

Given $f(x)=3\cdot2^{x}$:

• $f(-2)=3\cdot2^{-2}=\frac34$.

• $f(0)=3$.

• $f(2)=3\cdot4=12$.

Continuous Compounding

$A=Pe^{rt}$ (base $e\approx2.71828$).

Logarithmic Functions

• Inverse of exponential: $y=\log_b(x)\iff b^{y}=x$.

• Domain $(0,\infty)$; range $(-\infty,\infty)$.

• Vertical asymptote $x=0$.

• Passes through $(1,0)$.

• Laws
  • Product: $\log<em>b(MN)=\log</em>bM+\log<em>bN$. • Quotient: $\log</em>b\tfrac{M}{N}=\log<em>bM-\log</em>bN$.
  • Power: $\log<em>b(M^k)=k\,\log</em>bM$.
  • Change of base: $\log<em>bM=\dfrac{\log</em>kM}{\log_kb}$.

Solving Equations – Illustrations

1. Exponential: $5^{x}=625\Rightarrow5^{x}=5^{4}\Rightarrow x=4$.

2. Mixed: $(25)^{x+2}=5^{3x-4}$

   • Write $25=5^{2}$ → $5^{2x+4}=5^{3x-4}$ → $2x+4=3x-4$ → $x=8$.

3. Logarithmic: $2\log<em>7x=\log</em>781$ → $\log<em>7x^{2}=\log</em>781$ → $x^{2}=81\Rightarrow x=9$ (discard $-9$ by domain).

Applications

• Finance (compound interest): $B=50\,000(1.06)^{t}$; after $10$ y, $B\approx89\,542.38$.

• Population growth: $P(t)=P<em>0e^{kt}$; if $P(5)=2P</em>0$, then $k=\tfrac{\ln2}{5}$.

• pH scale: $\text{pH}=-\log_{10}[H^+]$. If $[H^+]=10^{-3}$ M ⇒ $\text{pH}=3$.

ASSESSMENT HIGHLIGHTS (MCQs)

Functions & Graphs

• Range of $f(x)=x^{2}+1$ is $[1,\infty)$.

• Domain of $g(x)=\sqrt{x}$ is $x\ge0$ (choice "a>0" equivalent).

• Relation $g(x)=\pm\sqrt{x}$ is not a function (fails VLT).

Rational Section

• Domain of $f(x)=\dfrac{2x+1}{x^{2}-4}$ excludes $x=\pm2$.

• Inequality \dfrac{3x}{x+1}>1 solved by sign analysis → (x<-1\,\text{or}\,x>2).

Exponential Section

• General form is $y=ab^{x}$.

• Evaluate $f(3)=2^{3}=8$.

• Graph $2^{x}$ always passes through $(0,1)$.

• Expression $y=2(0.5)^{x}$ represents decay (base <1).

• $e^{0}=1$.

• Continuous compounding: $A=1000e^{0.05t}$.

Logarithmic Section

• Solve $\log_{2}x=5$ → $x=32$.

• $\ln(e^{3})=3$.

• $y=\log_{3}x$ has vertical asymptote at $x=0$ and is not symmetric about $y$-axis.

REFERENCES & MULTIMEDIA LINKS

• Orlando A. O. et al., "E-Math Worktext in General Mathematics," 2nd Ed., Rex Education (2024).

• Renard E. L. C. et al., "Soaring 21st-Century Mathematics," Phoenix (2023).

• YouTube support

  • Polynomial Functions: https://www.youtube.com/watch?v=L-ta6mIJQ0Y

  • Vertical Line Test: https://www.youtube.com/watch?v=Mxe2lX1htNk

  • Graphing Rational Functions: https://www.youtube.com/watch?v=fy45qX8cUwQ

  • Exponential Functions: https://www.youtube.com/watch?v=nqpn0SQB5ds

  • Logarithmic Functions: https://www.youtube.com/watch?v=EvQCLCaiq2U

FUNCTIONS AND THEIR GRAPHS

Objectives

• Define function, domain, range.

• Use Vertical Line Test (VLT).

• Evaluate, graph, and interpret functions (linear & quadratic).

Definition of a Function

• A function ff assigns exactly one output f(x)f(x) to each input xx in its domain AA (f:A→Bf:A→B).

• Domain: Permissible xx–values.

• Range: Attainable f(x)f(x)–values.

Core Families of Algebraic Functions

• Polynomial: f(x)=anxn+⋯+a0f(x)=anxn+⋯+a0.

• Rational: f(x)=p(x)q(x),  q(x)≠0f(x)=q(x)p(x)​,q(x)=0.

• Root / Radical: f(x)=p(x)nf(x)=np(x)​.

• Implicit algebraic: Defined by an equation in xx & yy (e.g. x2+y2=9x2+y2=9).

Properties (illustrative)

• Domain restrictions: e.g., f(x)=x⇒x≥0f(x)=x​⇒x≥0.

• Behaviours:

  • Polynomials: Smooth, continuous.

  • Rationals: Discontinuities, asymptotes.

Vertical Line Test (VLT)

• "A graph represents a function iff no vertical line intersects it more than once."

• Examples:

  • Parabola y=x2y=x2 ✓

  • Circle x2+y2=9x2+y2=9 ✗

Evaluating Functions – Algorithm

1. Identify f(x)f(x).

2. Substitute input.

3. Simplify.

• Linear: f(x)=3x+5;  f(4)=17f(x)=3x+5;f(4)=17.

• Quadratic: g(x)=x2−4x+3;  g(−2)=15g(x)=x2−4x+3;g(−2)=15.

• Fractional: h(x)=x+1x−2;  h(3)=4h(x)=x−2x+1​;h(3)=4. Undefined at x=2x=2.

• Common Pitfalls: Dropping parentheses with negatives; forgetting zero denominators.

Graphing Basics

• Steps:

  1. Identify type & traits.

  2. Choose representative xx–values.

  3. Compute yy–values.

  4. Plot (x,y)(x,y) pairs.

  5. Connect smoothly.

• Key features: Intercepts, slope/vertex, asymptotes.

  • Linear Example: y=2x+1y=2x+1.

  • Quadratic Example: f(x)=x2−4x+3f(x)=x2−4x+3 (vertex at x=2x=2).

Real-World Application (T-Shirt Company)

• Cost: C(x)=0.20x2+5x+200C(x)=0.20x2+5x+200

• Revenue: R(x)=12xR(x)=12x

• Profit: P(x)=R(x)−C(x)=−0.20x2+7x−200P(x)=R(x)−C(x)=−0.20x2+7x−200

• Profit at 50 shirts: P(50)=−350P(50)=−350 (loss).

• Break-even: P(x)=0⇒x≈29P(x)=0⇒x≈29 shirts.

RATIONAL EQUATIONS, FUNCTIONS & INEQUALITIES

Objectives

• Solve rational equations/inequalities.

• Graph rational functions & locate asymptotes.

• State domain & range.

• Apply concepts to real-world problems.

Prerequisite Skills

• Fraction operations, factoring, solving linear/quadratic equations.

• Domain restrictions (denominator ≠0=0); checking extraneous roots.

Rational Equations – Prototypes & Methods

1. Simple: 3x=4⇒x=34x3​=4⇒x=43​.

2. Two terms: xx+2=34x+2x​=43​ (cross-multiply, solve quadratic).

3. Quadratic rational: x+1x−2=xx−2x+1​=x (cross-multiply).

4. Combined numerators: xx−1+2x−1=5x−1x​+x−12​=5.

5. Application (pipes): 14+(−16)=1t41​+(−61​)=t1​.

Worked Example (Intermediate)

• 1x+1x+2=34x1​+x+21​=43​

• LCD 4x(x+2)4x(x+2), leads to 3x2−2x−8=0⇒(3x+4)(x−2)=03x2−2x−8=0⇒(3x+4)(x−2)=0.

• Roots: x=−43x=−34​ (allowed), x=2x=2 (reject due to restriction).

Rational Functions – Anatomy

• General form f(x)=p(x)q(x),  q(x)≠0f(x)=q(x)p(x)​,q(x)=0.

• Domain: All real xx where q(x)≠0q(x)=0.

Graph Features

• Vertical asymptotes: Zeros of q(x)q(x) (after canceling holes).

• Horizontal asymptote: Compare degrees of pp & qq.

• Holes: Common factors canceled.

Sample Catalog

1. f(x)=1xf(x)=x1​: VA x=0x=0, HA y=0y=0.

2. g(x)=2x+1x−3g(x)=x−32x+1​: VA x=3x=3, HA y=2y=2.

3. h(x)=x2−1x−1=x+1h(x)=x−1x2−1​=x+1 with hole at x=1x=1.

Rational Inequalities – Strategy

1. Rewrite to zero on one side.

2. Factor numerator & denominator.

3. Mark critical points (zeros & undefineds) on number line.

4. Test intervals; respect inequality sign.

5. State solution set, exclude undefined points.

Example

• Solve x+2x−1≤0x−1x+2​≤0.

• Critical points x=−2,1x=−2,1. Test intervals.

• Solution: [−2,1)[−2,1).

Mixed Applied Problems

• Work: Amy (6h) & Ben (4h). Together: 1x=16+14⇒x=2.4 hx1​=61​+41​⇒x=2.4 h.

• Motion: 60 km downstream (3h), upstream (5h). Boat speed (b), Current speed (c): \begin{cases}b+c=20\b-c=12\end{cases} \Rightarrow b=16\text{ km/h},\;c=4\text{ km/h}.

EXPONENTIAL & LOGARITHMIC FUNCTIONS

Objectives

• Describe properties of axax & log⁡b(x)logb​(x).

• Solve related equations.

• Graph and identify asymptotes/intercepts.

• Apply models of growth, decay, pH, finance.

Essential Prior Knowledge

• Algebraic manipulation, exponent laws, function notation, inverses, transformations.

Exponential Functions

• General form f(x)=abxf(x)=abx (a≠0,  b>0,  b≠1a=0,b>0,b=1).

• Growth: b>1b>1; decay: 0<b<10<b<1.

• Domain (−∞,∞)(−∞,∞); range (0,∞)(0,∞).

• Horizontal asymptote y=0y=0.

• Key point (0,a)(0,a).

• Continuous Compounding: A=PertA=Pert.

Logarithmic Functions

• Inverse of exponential: y=log⁡b(x)  ⟺  by=xy=logb​(x)⟺by=x.

• Domain (0,∞)(0,∞); range (−∞,∞)(−∞,∞).

• Vertical asymptote x=0x=0.

• Passes through (1,0)(1,0).

• Laws:

  • Product: \logb(MN)=\logbM+\log_bN.

  • Quotient: \logb\tfrac{M}{N}=\logbM-\log_bN.

  • Power: \logb(M^k)=k\,\logbM.

  • Change of base: \logbM=\dfrac{\logkM}{\log_kb}.

Solving Equations – Illustrations

1. Exponential: 5x=625⇒x=45x=625⇒x=4.

2. Mixed: (25)x+2=53x−4⇒x=8(25)x+2=53x−4⇒x=8.

3. Logarithmic: 2log⁡7x=log⁡781⇒x=92log7x=log781⇒x=9.

Applications

• Finance: B=50 000(1.06)tB=50000(1.06)t; after 10y, B≈89 542.38B≈89542.38.

• Population growth: P(t)=P0ektP(t)=P0​ekt.

• pH scale: pH=−log⁡10[H+]pH=−log10​[H+].

ASSESSMENT HIGHLIGHTS (MCQs)

Functions & Graphs

• Range of f(x)=x2+1f(x)=x2+1 is [1,∞)[1,∞).

• Domain of g(x)=xg(x)=x​ is x≥0x≥0.

• Relation g(x)=±xg(x)=±x​ is not a function.

Rational Section

• Domain of f(x)=2x+1x2−4f(x)=x2−42x+1​ excludes x=±2x=±2.

• Inequality 3xx+1>1x+13x​>1 solved by sign analysis (x<−1 or x>2)(x<−1orx>2).

Exponential Section

• General form: y=abxy=abx.

• Evaluate f(3)=23=8f(3)=23=8.

• Graph 2x2x always passes through (0,1)(0,1).

• y=2(0.5)xy=2(0.5)x represents decay (b<1b<1).

• e0=1e0=1.

• Continuous compounding: A=1000e0.05tA=1000e0.05t.

Logarithmic Section

• Solve log⁡2x=5⇒x=32log2​x=5⇒x=32.

• ln⁡(e3)=3ln(e3)=3.

• y=log⁡3xy=log3​x has VA at x=0x=0, not symmetric about yy-axis.

REFERENCES & MULTIMEDIA LINKS

• Orlando A. O. et al., "E-Math Worktext in General Mathematics," 2nd Ed., Rex Education (2024).

• Renard E. L. C. et al., "Soaring