Sums & Difference of Functions

Add and subtract function expressions and graph results. Grade 12 Math 30-1.

Lesson 1.3 of Function Transformations & Operations in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Addition of Functions. Learn how to add two functions together and understand the notation (f + g)(x) for combining functions through addition.
• Subtraction of Functions. Learn how to subtract one function from another and understand the notation (f - g)(x) for combining functions through subtraction.

Inside the lesson — a free preview

What Does It Mean to Add Functions?

Starting Simple. We already know how to add numbers and algebraic expressions:

3 + 6a or (2x + 5) + (3x - 1) = 5x + 4

We can do the exact same thing with functions!

The Big Idea. If we have two functions f(x) and g(x), we can combine them by adding their outputs (the y-values) at the same x-value.

Sum of Functions

(f + g)(x) = f(x) + g(x)

In Plain English. To find (f + g)(x):

1. Take the first function f(x)

2. Add the second function g(x)

3. Simplify if possible

Adding Functions (Algebraically)

Example 1. Let's add two functions step by step.

Given:

f(x) = -4x - 3

g(x) = 2x^2

Find: (f + g)(x)

Solution. Step 1: Write the definition

(f + g)(x) = f(x) + g(x)

Step 2: Substitute the functions

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

Step 3: Rearrange in standard form (highest power first)

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

✅ Important Property. Is (f + g)(x) = (g + f)(x)?

YES! Addition is commutative — order does not matter.

This will always be true for addition.

Understanding the Domain of Addition

Important Concept. When adding functions, the domain of (f + g)(x) is the intersection of the domains of f(x) and g(x).

In Plain English. For (f + g)(x) to exist, BOTH f(x) and g(x) must be defined at that x-value.

Domain of (f + g)(x) = values where BOTH functions work

Example 3 - Finding the Domain. Given:

f(x) = √(x - 3)

g(x) = (1)/(x + 2)

Find: The domain of (f + g)(x)

Solution. Step 1: Find the domain of f(x) = √(x - 3)

For square roots, the radicand must be non-negative:

x - 3 ≥ 0

x ≥ 3

Domain of f: [3, ∞)

Step 2: Find the domain of g(x) = (1)/(x + 2)

For fractions, the denominator cannot be zero:

x + 2 ≠ 0

x ≠ -2

Domain of g: (-∞, -2) (-2, ∞) (all real numbers except -2)

Step 3: Find the intersection

We need values where BOTH functions work:

• f needs: x ≥ 3

• g needs: x ≠ -2

Since all values x ≥ 3 are already greater than -2, the restriction x ≠ -2 is automatically satisfied.

Domain of (f + g)(x): [3, ∞) or x ≥ 3

💡 Key Insight. The more restrictive domain becomes the domain of the sum.

In this case, x ≥ 3 is more restrictive than x ≠ -2, so it determines the final domain.

Worked example: Another Example

Example 2. Given:

f(x) = 3x + 7

g(x) = -2x^2 + 5x

Find: (f + g)(x)

Solution. Step 1: Write the definition

(f + g)(x) = f(x) + g(x)

Step 2: Substitute

(f + g)(x) = (3x + 7) + (-2x^2 + 5x)

Step 3: Combine like terms

(f + g)(x) = -2x^2 + 3x + 5x + 7

(f + g)(x) = -2x^2 + 8x + 7

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