Math 110 Readiness: Domain Restrictions, Parabola Vertex & Perpendicular Lines

Domain of Functions – General Principles

• In Math 110 most common elementary functions (polynomials, exponentials, trigonometric, etc.) have domain $(-\infty,\,\infty)$.
• Three major families force you to remove numbers from the real‐number line:
  1. Rational functions (division by zero).
  2. Even–indexed radicals (negative radicands).
  3. Logarithmic functions (non-positive arguments).
• Always check whether a function is piece-wise defined.
  • Restrictions that fall outside the stated sub-interval are ignored.
  • Only restrictions inside the sub-interval matter.

1 Rational Functions

• A rational function has the form $f(x)=\dfrac{p(x)}{g(x)}$, where $p(x),\,g(x)$ are polynomials.
• Rule: any $x$ that makes $g(x)=0$ must be excluded.
• Procedure:
  1. Factor the denominator $g(x)$ if possible.
  2. Solve $g(x)=0$.
  3. Remove those roots from $\mathbb R$.
• Example
  • $y=\dfrac{15}{x^{2}+2x}$
  • Denominator: $x^{2}+2x=x(x+2)=0$
  • Zeros: $x=0,\;x=-2$
  • Domain: $(-\infty,-2)\cup(-2,0)\cup(0,\infty)$.

2 Even-Indexed Radical Functions

• A radical with an even index (square root, fourth root, …) is defined only for non-negative radicands.
• Rule: set the radicand $\ge 0$.
• Procedure:
  1. Identify the radicand $r(x)$.
  2. Solve $r(x)\ge 0$.
  3. Resulting set is the domain.
• Example
  • $y = \sqrt{3x-4}$
  • Condition: $3x-4\ge0\;\Longrightarrow\;x\ge\tfrac43$
  • Domain: $[\tfrac43,\,\infty)$.
• Piece-wise note: if the same radical is declared only for x>1, then the final domain would be $[\tfrac43,\,\infty)$ intersected with $\,(1,\infty)$, which remains $[\tfrac43,\infty)$.

3 Logarithmic Functions

• A logarithm $y=\log_a\bigl(f(x)\bigr)$ (with a>0,\,a\ne1) exists only when its argument is strictly positive.
• Rule: set f(x)>0.
• Procedure mirrors the radical test but uses a strict inequality.
• Example
  • $y=\log_{10}(x^{2}-5x)$
  • Condition: x^{2}-5x>0 \;\Longrightarrow\; x

4 Quick Scan for No Domain Restrictions

• Linear functions $y=mx+b$ – always defined.
• Odd-indexed radicals $\sqrt[3]{\cdot}\,,$ $\sqrt[5]{\cdot}$ – the radicand may be negative.
• Polynomials – unlimited domain.

5 Vertex of a Parabola

Given the conic in implicit form
$x^{2}+y-2x+6=0,$
solve for $y$ to place it in standard quadratic form:
$y=-x^{2}+2x-6.$
Interpret as $y=ax^{2}+bx+c$ with parameters
$a=-1,\;b=2,\;c=-6.$

• Vertex shortcut (completing‐the‐square / calculus):
  • $h=-\dfrac{b}{2a}= -\dfrac{2}{2(-1)} = 1$
  • $k=f(h)=a h^{2}+b h + c = (-1)(1)^{2}+2(1)-6 = -1+2-6 = -5$
  • Vertex $V=(1,-5).$

6 Lines – Forms & Slopes

• Ax+By=C → slope $m=-\dfrac{A}{B}$ (provided $B\neq0$).
• Point–slope form: $y-y<em>{0}=m(x-x</em>{0}).$
• Perpendicular slopes: if $m<em>{1}m</em>{2}=-1$, the two lines are perpendicular.

7 Example – Line Through a Vertex, Perpendicular to a Given Line

Task: “Find the equation of the line that passes through the vertex of the parabola $x^{2}+y-2x+6=0$ and is perpendicular to $3x-2y=2.$”

1. Vertex (from Section 5): $V(1,-5).$
2. Slope of given line: rewrite $3x-2y=2\Rightarrow y=\tfrac32x-1,$ so $m_{1}=\tfrac32.$
3. Slope of requested line: $m<em>{2}=-\dfrac{1}{m</em>{1}}=-\dfrac23.$
4. Point–slope equation:
   $y+5=-\frac23\,(x-1).$
5. Standard form (multiply by $3$):
   $3(y+5) = -2(x-1)\;\Longrightarrow\;3y+15 = -2x+2.$
6. Rearrange:
   $2x +3y +13 = 0.$
7. Answer set (all equivalent):
   • $y+5=-\dfrac23(x-1)$
   • $3y+15=-2x+2$
   • $2x+3y+13=0.$
   Each represents the same perpendicular line.

8 Connections & Remarks

• The domain tests for radicals and logs mirror each other; logs are strict (>0), radicals are non-strict (≥0).
• Piece-wise definitions often appear on exams; always perform interval intersection between the intrinsic restriction and the stated sub-interval.
• Vertex calculation is fundamental for graph-based questions, maximum/minimum problems, and tangent line tasks.
• The perpendicular‐line skill is frequently paired with distance‐to‐a‐line or intersection problems; storing the slope-relationship $m<em>{1}m</em>{2}=-1$ is critical.

9 Quick Reference – Formulas

• Denominator ≠ 0: $g(x)\neq0.$
• Even radical: $r(x)\ge0.$
• Logarithm: f(x)>0.
• Vertex $h=-\dfrac{b}{2a},\quad k=f(h).$
• Point–slope $y-y<em>{0}=m(x-x</em>{0}).$
• Perpendicular $m<em>{2}=-\dfrac1{m</em>{1}}.$

10 Ethical & Practical Takeaways

• Clearly stating domains prevents misuse of formulas in modeling and ensures that numeric methods (e.g.
  calculators, computer algebra systems) do not produce undefined operations such as division by zero or taking square roots of negative numbers.
• In real-world contexts (engineering tolerances, population models, finance), these restrictions translate into feasible vs. infeasible parameter ranges.