Lesson 3: Function Arithmetic
Estimated time: 25-30 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Add, subtract, multiply, and divide functions
- Determine the domain of combined functions
- Evaluate combined functions at specific values
- Understand domain restrictions from division
Combining Functions
Just like we can add, subtract, multiply, and divide numbers, we can perform these same operations on functions! This is called function arithmetic.
The idea is simple: to combine two functions, you perform the operation on their outputs (the function values).
Function Arithmetic: Given two functions f(x) and g(x), we can create new functions by combining them:
- Addition: (f + g)(x) = f(x) + g(x)
- Subtraction: (f − g)(x) = f(x) − g(x)
- Multiplication: (f · g)(x) = f(x) · g(x)
- Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0
The Four Operations Explained
Let's work through each operation with the same two functions to see how they work:
f(x) = x² and g(x) = 3x
1. Addition: (f + g)(x)
Example:
(f + g)(x) = x² + 3x
Evaluate at x = 2:
(f + g)(2) = f(2) + g(2)
= (2)² + 3(2)
= 4 + 6
= 10
2. Subtraction: (f − g)(x)
Example:
(f − g)(x) = x² − 3x
Evaluate at x = 2:
(f − g)(2) = f(2) − g(2)
= (2)² − 3(2)
= 4 − 6
= −2
Important: Order matters! (f − g)(x) ≠ (g − f)(x)
3. Multiplication: (f · g)(x)
Example:
(f · g)(x) = x² · 3x = 3x³
Evaluate at x = 2:
(f · g)(2) = f(2) · g(2)
= (2)² · 3(2)
= 4 · 6
= 24
4. Division: (f / g)(x)
Example:
(f / g)(x) = x² / 3x = x/3 (for x ≠ 0)
Evaluate at x = 2:
(f / g)(2) = f(2) / g(2)
= (2)² / 3(2)
= 4 / 6
= 2/3 or approximately 0.667
Critical: We must exclude values where g(x) = 0 (division by zero!)
Step-by-Step Example
Example: Complete Function Arithmetic
Given f(x) = 2x + 1 and g(x) = x² − 3, find:
a) (f + g)(x)
(f + g)(x) = f(x) + g(x)
= (2x + 1) + (x² − 3)
= x² + 2x + 1 − 3
= x² + 2x − 2
b) (f − g)(x)
(f − g)(x) = f(x) − g(x)
= (2x + 1) − (x² − 3)
= 2x + 1 − x² + 3
= −x² + 2x + 4
Note: Be careful with the negative sign! Distribute it across (x² − 3)
c) (f · g)(x)
(f · g)(x) = f(x) · g(x)
= (2x + 1)(x² − 3)
= 2x(x²) + 2x(−3) + 1(x²) + 1(−3)
= 2x³ − 6x + x² − 3
= 2x³ + x² − 6x − 3
d) (f / g)(x)
(f / g)(x) = f(x) / g(x)
= (2x + 1) / (x² − 3)
= (2x + 1) / (x² − 3)
Domain restriction: x² − 3 ≠ 0
x² ≠ 3
x ≠ ±√3
So x ≠ √3 and x ≠ −√3
Domain of Combined Functions
When we combine functions, we need to carefully consider the domain of the resulting function.
Domain Rules for Combined Functions:
- For (f + g)(x), (f − g)(x), and (f · g)(x):
Domain is the intersection of the domains of f and g
(Both functions must be defined) - For (f / g)(x):
Domain is the intersection of the domains of f and g,
EXCLUDING any values where g(x) = 0
Example: Finding Domain of Combined Functions
Given: f(x) = √x (domain: [0, ∞)) and g(x) = x − 4 (domain: all real numbers)
a) Domain of (f + g)(x):
We need BOTH f(x) and g(x) to be defined
Domain of f: [0, ∞)
Domain of g: all real numbers
Intersection: [0, ∞)
b) Domain of (f / g)(x):
Start with intersection from part a): [0, ∞)
Additionally, exclude where g(x) = 0:
x − 4 = 0 → x = 4
Domain: [0, 4) ∪ (4, ∞)
(All non-negative numbers except 4)
Example: Another Domain Problem
Given: f(x) = 1/x (domain: x ≠ 0) and g(x) = √(x − 1) (domain: [1, ∞))
Find domain of (f · g)(x):
Domain of f: all real numbers except 0
Domain of g: [1, ∞)
Intersection: [1, ∞)
Note: x = 0 is already excluded by g's domain, so we just take [1, ∞)
Common Mistakes to Avoid
Mistake #1: Forgetting to Distribute Negatives
Wrong: (f − g)(x) where f(x) = x and g(x) = x² − 3
x − x² − 3 (forgot to distribute the negative!)
Correct:
x − (x² − 3) = x − x² + 3
Mistake #2: Ignoring Division by Zero
Wrong: Saying (f / g)(x) = (x + 1)/(x − 2) has domain: all real numbers
Correct:
Domain: All real numbers except x = 2 (where denominator = 0)
Mistake #3: Thinking (f − g) and (g − f) are the Same
Order matters in subtraction!
If f(x) = 5x and g(x) = 2x:
- (f − g)(x) = 5x − 2x = 3x
- (g − f)(x) = 2x − 5x = −3x
- These are different!
Mistake #4: Not Considering Both Domains
When combining f(x) = √x and g(x) = 1/(x−1), you must consider:
- f requires x ≥ 0
- g requires x ≠ 1
- Combined: x ≥ 0 AND x ≠ 1
- Final domain: [0, 1) ∪ (1, ∞)
Check Your Understanding
Try these questions to see if you've grasped the key concepts:
1. Given f(x) = 2x and g(x) = x + 5, find (f + g)(3).
(f + g)(3) = f(3) + g(3)
= 2(3) + (3 + 5)
= 6 + 8
= 14
2. Given f(x) = x² and g(x) = 4x, find (f · g)(x) in simplified form.
(f · g)(x) = f(x) · g(x)
= x² · 4x
= 4x³
3. Given f(x) = x + 2 and g(x) = x − 3, find (f / g)(x) and state its domain.
(f / g)(x) = (x + 2) / (x − 3)
Domain: We must exclude where g(x) = 0:
x − 3 = 0
x = 3
Domain: All real numbers except x = 3 (or (−∞, 3) ∪ (3, ∞))
Key Takeaways
- (f + g)(x) = f(x) + g(x) — Add the function values
- (f − g)(x) = f(x) − g(x) — Subtract the function values (order matters!)
- (f · g)(x) = f(x) · g(x) — Multiply the function values
- (f / g)(x) = f(x) / g(x) — Divide the function values (g(x) ≠ 0)
- For +, −, ×: Domain is the intersection of individual domains
- For ÷: Domain is the intersection, EXCLUDING where the denominator equals zero
- Always simplify your final answer and clearly state any domain restrictions
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