Function Operations
📌 What are Function Operations?
Just like numbers, functions can be combined using arithmetic operations and composition to create new functions. There are two main categories:
- Arithmetic Operations: Addition, Subtraction, Multiplication, Division
- Composition of Functions: Using the output of one function as the input of another
Addition of Functions
Definition:
\( (f + g)(x) = f(x) + g(x) \)
Add the outputs of both functions for the same input \(x\)
📝 Examples:
Example 1: Given \( f(x) = 3x + 2 \) and \( g(x) = x - 5 \), find \( (f + g)(x) \)
\( (f + g)(x) = f(x) + g(x) \)
\( = (3x + 2) + (x - 5) \)
\( = 4x - 3 \)
Example 2: Given \( f(x) = x^2 + 3x \) and \( g(x) = 2x + 1 \), find \( (f + g)(x) \)
\( (f + g)(x) = (x^2 + 3x) + (2x + 1) \)
\( = x^2 + 5x + 1 \)
Example 3: Evaluate \( (f + g)(2) \) if \( f(x) = 2x - 1 \) and \( g(x) = x^2 \)
Method 1: Find the sum first
\( (f + g)(x) = 2x - 1 + x^2 = x^2 + 2x - 1 \)
\( (f + g)(2) = 2^2 + 2(2) - 1 = 4 + 4 - 1 = 7 \)
Method 2: Evaluate each function then add
\( f(2) = 2(2) - 1 = 3 \)
\( g(2) = 2^2 = 4 \)
\( (f + g)(2) = 3 + 4 = 7 \)
Subtraction of Functions
Definition:
\( (f - g)(x) = f(x) - g(x) \)
Subtract the second function from the first
📝 Examples:
Example 1: Given \( f(x) = 5x + 7 \) and \( g(x) = 2x - 3 \), find \( (f - g)(x) \)
\( (f - g)(x) = f(x) - g(x) \)
\( = (5x + 7) - (2x - 3) \)
\( = 5x + 7 - 2x + 3 \)
\( = 3x + 10 \)
Example 2: Given \( f(x) = x^2 + 2x - 1 \) and \( g(x) = 3x + 4 \), find \( (f - g)(x) \)
\( (f - g)(x) = (x^2 + 2x - 1) - (3x + 4) \)
\( = x^2 + 2x - 1 - 3x - 4 \)
\( = x^2 - x - 5 \)
⚠️ Important: Be careful with negative signs when distributing!
Multiplication of Functions
Definition:
\( (f \cdot g)(x) = f(x) \cdot g(x) \)
Multiply the outputs of both functions
📝 Examples:
Example 1: Given \( f(x) = 2x \) and \( g(x) = x + 3 \), find \( (f \cdot g)(x) \)
\( (f \cdot g)(x) = f(x) \cdot g(x) \)
\( = (2x)(x + 3) \)
\( = 2x^2 + 6x \)
Example 2: Given \( f(x) = x - 2 \) and \( g(x) = x + 5 \), find \( (f \cdot g)(x) \)
\( (f \cdot g)(x) = (x - 2)(x + 5) \)
Using FOIL:
\( = x^2 + 5x - 2x - 10 \)
\( = x^2 + 3x - 10 \)
Example 3: Given \( f(x) = x^2 - 1 \) and \( g(x) = 3x \), find \( (f \cdot g)(x) \)
\( (f \cdot g)(x) = (x^2 - 1)(3x) \)
\( = 3x^3 - 3x \)
Division of Functions
Definition:
\( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \) where \( g(x) \neq 0 \)
Divide the first function by the second
⚠️ The denominator cannot be zero!
📝 Examples:
Example 1: Given \( f(x) = 6x^2 \) and \( g(x) = 3x \), find \( \left(\frac{f}{g}\right)(x) \)
\( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{6x^2}{3x} = 2x \)
Restriction: \( x \neq 0 \) (because \( g(x) \neq 0 \))
Example 2: Given \( f(x) = x^2 - 4 \) and \( g(x) = x - 2 \), find \( \left(\frac{f}{g}\right)(x) \)
\( \left(\frac{f}{g}\right)(x) = \frac{x^2 - 4}{x - 2} \)
Factor the numerator:
\( = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \)
Restriction: \( x \neq 2 \)
Example 3: Given \( f(x) = x^2 + 3x - 10 \) and \( g(x) = x + 5 \), find \( \left(\frac{f}{g}\right)(x) \)
\( \left(\frac{f}{g}\right)(x) = \frac{x^2 + 3x - 10}{x + 5} \)
Factor: \( x^2 + 3x - 10 = (x + 5)(x - 2) \)
\( = \frac{(x + 5)(x - 2)}{x + 5} = x - 2 \)
Restriction: \( x \neq -5 \)
Domain of Function Operations
Important Rules:
For \( f + g \), \( f - g \), and \( f \cdot g \):
Domain = Domain of \(f\) ∩ Domain of \(g\)
(Intersection: values that work for BOTH functions)
For \( \frac{f}{g} \):
Domain = (Domain of \(f\) ∩ Domain of \(g\)) excluding where \(g(x) = 0\)
Composition of Functions
What is Composition?
Composition of functions means using the output of one function as the input of another function.
\( (f \circ g)(x) = f(g(x)) \)
Read as "f of g of x" or "f composed with g"
How it works:
- Step 1: Start with the inside function \(g(x)\)
- Step 2: Evaluate \(g(x)\) to get a result
- Step 3: Use that result as the input for \(f(x)\)
- Step 4: Evaluate \(f\) at that input
⚠️ Important: \( f \circ g \neq g \circ f \) (Order matters!)
Composition of Linear Functions
Key Fact:
The composition of two linear functions is also a linear function!
If \( f(x) = ax + b \) and \( g(x) = cx + d \)
Then \( (f \circ g)(x) = a(cx + d) + b = acx + ad + b \)
📝 Example 1 - Find a Value:
Given \( f(x) = 2x + 3 \) and \( g(x) = 4x - 1 \), find \( f(g(2)) \)
Step 1: Find \( g(2) \)
\( g(2) = 4(2) - 1 = 8 - 1 = 7 \)
Step 2: Find \( f(7) \)
\( f(7) = 2(7) + 3 = 14 + 3 = 17 \)
Answer: \( f(g(2)) = 17 \)
📝 Example 2 - Find an Equation:
Given \( f(x) = 3x - 5 \) and \( g(x) = 2x + 1 \), find \( (f \circ g)(x) \)
Step 1: Write as \( f(g(x)) \)
Step 2: Replace \( g(x) \) with its expression
\( f(g(x)) = f(2x + 1) \)
Step 3: Substitute \( (2x + 1) \) into \( f(x) \) wherever there's an \( x \)
\( f(2x + 1) = 3(2x + 1) - 5 \)
\( = 6x + 3 - 5 \)
\( = 6x - 2 \)
Answer: \( (f \circ g)(x) = 6x - 2 \)
📝 Example 3 - Both Directions:
Given \( f(x) = 5x + 2 \) and \( g(x) = x - 3 \), find both \( f(g(x)) \) and \( g(f(x)) \)
Find \( f(g(x)) \):
\( f(g(x)) = f(x - 3) \)
\( = 5(x - 3) + 2 \)
\( = 5x - 15 + 2 \)
\( = 5x - 13 \)
Find \( g(f(x)) \):
\( g(f(x)) = g(5x + 2) \)
\( = (5x + 2) - 3 \)
\( = 5x - 1 \)
Notice: \( f(g(x)) \neq g(f(x)) \) → Order matters!
Composition of Linear and Quadratic Functions
Key Facts:
- If you compose a linear function into a quadratic function, the result is quadratic
- If you compose a quadratic function into a linear function, the result is quadratic
- The degree of the composition is determined by the outside function
📝 Example 1 - Find a Value (Linear into Quadratic):
Given \( f(x) = x^2 + 2x \) and \( g(x) = 3x - 1 \), find \( f(g(1)) \)
Step 1: Find \( g(1) \)
\( g(1) = 3(1) - 1 = 2 \)
Step 2: Find \( f(2) \)
\( f(2) = 2^2 + 2(2) = 4 + 4 = 8 \)
Answer: \( f(g(1)) = 8 \)
📝 Example 2 - Find an Equation (Linear into Quadratic):
Given \( f(x) = x^2 - 4 \) and \( g(x) = 2x + 3 \), find \( f(g(x)) \)
Step 1: Substitute \( g(x) \) into \( f(x) \)
\( f(g(x)) = f(2x + 3) \)
Step 2: Replace \( x \) in \( f(x) \) with \( (2x + 3) \)
\( f(2x + 3) = (2x + 3)^2 - 4 \)
\( = (2x + 3)(2x + 3) - 4 \)
\( = 4x^2 + 6x + 6x + 9 - 4 \)
\( = 4x^2 + 12x + 5 \)
Answer: \( f(g(x)) = 4x^2 + 12x + 5 \)
📝 Example 3 - Find an Equation (Quadratic into Linear):
Given \( f(x) = 3x - 2 \) and \( g(x) = x^2 + 1 \), find \( f(g(x)) \)
Step 1: Substitute \( g(x) \) into \( f(x) \)
\( f(g(x)) = f(x^2 + 1) \)
Step 2: Replace \( x \) in \( f(x) \) with \( (x^2 + 1) \)
\( f(x^2 + 1) = 3(x^2 + 1) - 2 \)
\( = 3x^2 + 3 - 2 \)
\( = 3x^2 + 1 \)
Answer: \( f(g(x)) = 3x^2 + 1 \)
📝 Example 4 - Complete Problem:
Given \( f(x) = x^2 - 3x + 2 \) and \( g(x) = 2x + 1 \), find \( g(f(x)) \)
\( g(f(x)) = g(x^2 - 3x + 2) \)
\( = 2(x^2 - 3x + 2) + 1 \)
\( = 2x^2 - 6x + 4 + 1 \)
\( = 2x^2 - 6x + 5 \)
Answer: \( g(f(x)) = 2x^2 - 6x + 5 \)
⚡ Quick Summary
- Addition: \( (f + g)(x) = f(x) + g(x) \)
- Subtraction: \( (f - g)(x) = f(x) - g(x) \)
- Multiplication: \( (f \cdot g)(x) = f(x) \cdot g(x) \)
- Division: \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \), where \( g(x) \neq 0 \)
- Composition: \( (f \circ g)(x) = f(g(x)) \) (inside function first)
- For composition, work from the inside out
- Order matters in composition: \( f \circ g \neq g \circ f \)
- Domain of division excludes where denominator = 0
- Composition of two linear functions → linear function
- Composition involving quadratic → quadratic function
📚 Key Formulas Reference
Addition:
\( (f + g)(x) = f(x) + g(x) \)
Subtraction:
\( (f - g)(x) = f(x) - g(x) \)
Multiplication:
\( (f \cdot g)(x) = f(x) \cdot g(x) \)
Division:
\( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad g(x) \neq 0 \)
Composition (f of g):
\( (f \circ g)(x) = f(g(x)) \)
Composition (g of f):
\( (g \circ f)(x) = g(f(x)) \)
Domain for Operations:
Domain = (Domain of f) ∩ (Domain of g)