9–12 Math: Worked Micro-Examples

A) Functions & Models

1) Domain/Range (radical + rational)
Q: f(x)=5−2xx−3f(x)=\dfrac{\sqrt{5-2x}}{x-3}f(x)=x−35−2x​​. Find domain.
Steps:

1. Inside root ≥0 → 5−2x≥0⇒x≤2.55-2x\ge0\Rightarrow x\le 2.55−2x≥0⇒x≤2.5.
2. Denominator ≠0 → x≠3x\ne 3x=3 (already excluded by x≤2.5x\le2.5x≤2.5).
   Ans: (−∞, 2.5](-\infty,\,2.5](−∞,2.5].
   Pitfall: Forgetting both constraints (root & denom).

2) Transformations
Q: From y=f(x)y=f(x)y=f(x) to y=−2f(x+1)+3y=-2f(x+1)+3y=−2f(x+1)+3: describe.
Steps: Left 1; vertical stretch by 2; reflect across xxx-axis; up 3.
Pitfall: Getting the inside shift direction wrong.

3) Inverse existence
Q: Is f(x)=x2−4xf(x)=x^2-4xf(x)=x2−4x one-to-one on (−∞,2](-\infty,2](−∞,2]?
Steps: Complete square → (x−2)2−4(x-2)^2-4(x−2)2−4. On (−∞,2](-\infty,2](−∞,2] it’s decreasing ⇒ one-to-one ⇒ inverse exists.
Pitfall: Testing one-to-one on entire R\mathbb RR instead of restricted domain.

B) Systems

1) Method choice
Q: Solve {3x−2y=76x−4y=10\begin{cases}3x-2y=7\\ 6x-4y=10\end{cases}{3x−2y=76x−4y=10​.
Steps: Elimination shows LHS doubles, RHS doesn’t → inconsistent.
Ans: No solution (parallel lines).
Pitfall: Forcing substitution and missing inconsistency.

2) Word → system
Q: Two plans: $40 \$40$40 setup + 101010/hr vs $25 \$25$25 setup + 121212/hr. When equal?
Steps: 40+10h=25+12h⇒h=7.540+10h=25+12h\Rightarrow h=7.540+10h=25+12h⇒h=7.5 hrs.
Pitfall: Mixing setup fee into hourly rate.

C) Quadratics

1) Discriminant
Q: 2x2+3x+5=02x^2+3x+5=02x2+3x+5=0: root nature?
Steps: D=b2−4ac=9−40=−31<0⇒D=b^2-4ac=9-40=-31<0\RightarrowD=b2−4ac=9−40=−31<0⇒ two complex, non-real.
Pitfall: Solving fully when only nature is asked.

2) Vertex form
Q: y=3x2−12x+7y=3x^2-12x+7y=3x2−12x+7 → vertex & min.
Steps: y=3(x2−4x)+7=3[(x−2)2−4]+7=3(x−2)2−12+7=3(x−2)2−5y=3(x^2-4x)+7=3[(x-2)^2-4]+7=3(x-2)^2-12+7=3(x-2)^2-5y=3(x2−4x)+7=3[(x−2)2−4]+7=3(x−2)2−12+7=3(x−2)2−5.
Vertex (2,−5)(2,-5)(2,−5), min y=−5y=-5y=−5.
Pitfall: Forget to factor leading 3 before completing the square.

D) Exponentials & Logs

1) Solve
Q: 52x−1=75^{2x-1}=752x−1=7.
Steps: (2x−1)ln⁡5=ln⁡7⇒x=ln⁡7/ln⁡5+12(2x-1)\ln5=\ln7\Rightarrow x=\dfrac{\ln7/\ln5+1}{2}(2x−1)ln5=ln7⇒x=2ln7/ln5+1​.
Pitfall: Taking ln⁡(a+b)\ln(a+b)ln(a+b) as ln⁡a+ln⁡b\ln a+\ln blna+lnb (never!).

2) Growth model
Q: A=1200(1.04)tA=1200(1.04)^tA=1200(1.04)t. Double time?
Steps: 2=(1.04)t⇒t=ln⁡2/ln⁡1.04≈17.72= (1.04)^t \Rightarrow t=\ln2/\ln1.04\approx17.72=(1.04)t⇒t=ln2/ln1.04≈17.7 years.
Pitfall: Using 0.04t0.04t0.04t linear instead of exponential.

E) Trigonometry

1) Unit circle sign sanity
Q: sin⁡(210∘)\sin(210^\circ)sin(210∘), cos⁡(−π/3)\cos(-\pi/3)cos(−π/3).
Steps: 210∘210^\circ210∘ = QIII, ref 30∘30^\circ30∘ → −12-\tfrac12−21​.
cos⁡(−π/3)=cos⁡(π/3)=12\cos(-\pi/3)=\cos(\pi/3)=\tfrac12cos(−π/3)=cos(π/3)=21​.
Pitfall: Ignoring quadrant signs / mixing degree–radian.

2) Right-triangle solve
Q: θ=37∘ \theta=37^\circθ=37∘, hyp =10=10=10. Find opp, adj.
Steps: sin⁡θ=opp/10⇒opp≈10⋅0.601=6.01\sin\theta=\text{opp}/10\Rightarrow \text{opp}\approx10\cdot0.601=6.01sinθ=opp/10⇒opp≈10⋅0.601=6.01; cos⁡θ≈0.799⇒adj=7.99\cos\theta\approx0.799\Rightarrow \text{adj}=7.99cosθ≈0.799⇒adj=7.99.
Pitfall: Using wrong side labels (opp/adj/hyp).

F) Geometry / Proof / Similarity

1) Similar triangles scale
Q: △ABC∼△DEF\triangle ABC\sim \triangle DEF△ABC∼△DEF, AB:DE=3:5AB:DE=3:5AB:DE=3:5. If Area(ABC)=27 \text{Area}(ABC)=27Area(ABC)=27, find Area(DEF)\text{Area}(DEF)Area(DEF).
Steps: Side scale k=5/3k=5/3k=5/3 ⇒ area scale k2=25/9k^2=25/9k2=25/9. So 27⋅25/9=7527\cdot25/9=7527⋅25/9=75.
Pitfall: Scaling area linearly instead of quadratically.

2) Perpendicular lines in coordinate plane
Q: Line L1:3x−2y=7L_1: 3x-2y=7L1​:3x−2y=7. Find slope of lines ⟂ L1L_1L1​.
Steps: y=32x−72⇒m1=32y=\tfrac{3}{2}x-\tfrac{7}{2}\Rightarrow m_1=\tfrac{3}{2}y=23​x−27​⇒m1​=23​. Perp slope m2=−23m_2=-\tfrac{2}{3}m2​=−32​.
Pitfall: Taking negative instead of negative reciprocal.

G) Statistics & Probability

1) Conditional vs independence
Q: If P(A)=0.4,P(B)=0.5,P(A∩B)=0.2P(A)=0.4, P(B)=0.5, P(A\cap B)=0.2P(A)=0.4,P(B)=0.5,P(A∩B)=0.2, are A,BA,BA,B independent?
Steps: P(A)P(B)=0.2P(A)P(B)=0.2P(A)P(B)=0.2=P(A∩B)P(A\cap B)P(A∩B) ⇒ independent.
Pitfall: Checking P(A∣B)=P(B∣A)P(A|B)=P(B|A)P(A∣B)=P(B∣A) (not the test).

2) Sampling bias
Q: Online poll on a tech site: “Do you like VR?” Interpret.
Steps: Convenience/volunteer bias; not representative.
Pitfall: Treating any large nnn as unbiased.