Addition and Subtraction of Functions – Lesson 2.1

Introduction & Context

• Mathematics pervades daily life: from household appliances (washing machines, toaster ovens) to high-tech settings (computers, aircraft, laboratories).
• Unit focus: Operations on Functions (addition, subtraction, multiplication, division, composition).
• Lesson 2.1 concentrates on addition and subtraction of functions.
• DepEd competencies targeted (M11GM-Ia-3 & M11GM-Ia-4):
  • Perform the five operations on functions.
  • Solve real-world problems using functions.
• Specific lesson objectives:
  • Perform addition & subtraction of functions.
  • Apply these operations to problem solving.

Prerequisite Skills Review

• Students should be comfortable with:
  • Combining like terms, distributing constants, and managing negative signs.
  • Evaluating functions at specific values of $x$.
• Sample prerequisite tasks (show all work):
  • Simplify algebraic sums/differences such as $(2x^{2}+x-1)+(-7x+4)$.
  • Scalar multiplication & combination e.g. $3(4x-5y+2)+2(y+7x-8)$.
  • Evaluate functions, e.g. $f(x)=x^{2}-4x+5$ at $x=-1$, $g(x)=x^{4}-4x^{2}-x+5$ at $x=-2$, etc.

Warm-Up Activity (“Let’s Roll!”)

• Materials: colored papers with expressions (labelled 1–6), pair of dice, cartolina, marker.
• Procedure summary:
  • Roll the dice 5×; record the two corresponding expressions each time.
  • Roll a single die for each pair: odd → add, even → subtract.
  • Execute the indicated operation; write full solution; volunteers explain thinking aloud.
• Pedagogical goal: kinesthetic reinforcement that algebraic expressions (functions) can be added/subtracted.

Core Concepts: Sum & Difference of Functions

• Definition of a function: special relation assigning each input exactly one output (numerical, algebraic, real-life models).
• Sum of Functions: Given $f(x)$ and $g(x)$,
  $(f+g)(x)=f(x)+g(x).$
• Difference of Functions:
  $(f-g)(x)=f(x)-g(x).$
• Technique reminders:
  • When subtracting, rewrite as addition of opposites; change every sign in the subtrahend.
  • Combine like terms after grouping.

Worked Examples

• Example (basic polynomial addition)

• Given $f(x)=5x+2$ and $g(x)=8+6x-2x^{2}$,
  $(f+g)(x)=(5x+2)+(8+6x-2x^{2})$
  $=(2+8)+(5x+6x)-2x^{2}$
  $=10+11x-2x^{2}$
  (often reordered $=-2x^{2}+11x+10$).

• Example (basic polynomial subtraction)

• Same $f, g$ as above:
  $(f-g)(x)=(5x+2)-(8+6x-2x^{2})$
  $=(5x+2)+(-8-6x+2x^{2})$
  $=(2-8)+(5x-6x)+2x^{2}$
  $=-6-x+2x^{2}=2x^{2}-x-6.$

• Example 1 (addition with distribution)

• $g(x)=8+6x-2x^{2},\ h(x)=4(-x^{4}+x-3x^{2})$.
• Distribute 4 inside $h$: $h(x)=-4x^{4}+4x-12x^{2}$.
• Add & collect: $(g+h)(x)=-4x^{4}-14x^{2}+10x+8.$

• Example 2 (numeric evaluation after subtraction)

• $f(x)=5x+2,\ g(x)=15x+6$.
• Difference: $(f-g)(x)= -10x-4$.
• Evaluate at $x=4$: $(f-g)(4)=-10(4)-4=-44$.

• Example 3 (rational expressions)

• $f(x)=\dfrac{x+7}{x+1},\ g(x)=\dfrac{x-2}{x+3}$.
• LCD $(x+1)(x+3)=x^{2}+4x+3$.
• Numerator:
  $(x+7)(x+3)+(x-2)(x+1)$
  $=x^{2}+10x+21+x^{2}-x-2=2x^{2}+9x+19.$
• Result: $(f+g)(x)=\dfrac{2x^{2}+9x+19}{x^{2}+4x+3}.$

Real-World Applications

• Example 4 (salary model)

• Base salary: $f(x)=100x^{2}+2000x+150$ pesos.
• Incentive: $g(x)=20x^{2}+300x-500$ pesos.
• New salary (after quota met):
  $(f+g)(x)=120x^{2}+2300x-350$ pesos per month.

• Example 5 (wood cutting)

• Original length: $f(x)=9x^{2}-3x+2$ ft.
• Removed portion: $g(x)=2x^{2}+5x-4$ ft.
• Remaining length:
  $(f-g)(x)=7x^{2}-8x+6$ ft.

“Try It Yourself” Summaries (problems left for practice)

• Add $g(x)=x^{3}-x^{4}+6x-3$ and $h(x)=2(-x^{4}-2x+4x^{2})$.
• Compute $(f-g)(3)$ where $f(x)=5x^{2}-3,\ g(x)=x^{2}-15x+6$.
• Find $(f-g)(x)$ for the rational pair in Example 3.
• Salary problem with parameters: $f(x)=150x^{2}+1500x+250,\ g(x)=30x^{2}+500x-750$.
• Finance sheet: debit $d(x)=x^{5}+4x^{4}$, credit $c(x)=x^{3}+3-!x^{2}+x^{3}$; evaluate $(d-c)(5).$

Check Your Understanding Overview

• Set 1: Polynomial & rational combinations for functions $a(x)\text{ to }i(x)$; tasks include:
  • Symbolic sums/differences $(d+e)(x), (e+a)(x), (f-e)(x), (g+h)(x)$ etc.
  • Numeric substitutions, e.g. $(d+f)(-3), (b+c)(2), (i-a)(1).$
• Set 2: Express target functions $f<em>1\text{–}f</em>5$ as sum/difference of given set $p, q, r, s, t, u, v.$
• Set 3: Ingredient mixture word problem – find $m(x)=f(x)+i(x)=\dfrac{x+3}{x-5}+x^{2}-25.$

Key Points Recap

• Addition rule: $(f+g)(x)=f(x)+g(x).$
• Subtraction rule: $(f-g)(x)=f(x)-g(x).$
  • Always flip signs of subtrahend before combining.
• Like-term combination and distribution are fundamental skills.
• For rational functions: secure a common denominator (LCD) first.
• After performing operation, answers may be simplified or reordered (descending degree).

Connections & Implications

• Operations on functions mirror operations on real numbers & polynomials, enabling:
  • Construction of new models from simpler component models.
  • Modular analysis in physics, engineering, economics (salary incentives, material use, budgeting).
• Mastery lays groundwork for forthcoming topics: multiplication/division of functions, composition, and inverse functions.
• Ethical/practical note: Real-world salary incentives highlight importance of transparent mathematical modelling in HR & finance.

References (as provided)

• Pearson Education, “Function Operations and Composition.”
• Math is Fun, “Functions.”
• Math Motivation, “The Algebra of Functions.”