Subtracting Functions Algebraically- Methods and Examples
What Is Function Subtraction?
Subtracting functions means taking two functions and finding the difference between their outputs. You write it like this: (f - g)(x) = f(x) - g(x). That's the whole idea.
You can subtract any types of functions—polynomials, rational functions, square roots, whatever. The process stays the same. Combine like terms. Simplify. Done.
The Basic Method
Here's how you subtract functions algebraically:
- Write out both functions clearly
- Place them in parentheses with a minus sign between
- Distribute the negative sign to the second function
- Combine like terms
- Simplify the result
That third step trips up most people. When you subtract g(x), you need to subtract every term in g(x). Missing that negative sign on a term is how you get wrong answers.
Examples That Actually Help
Example 1: Polynomial Subtraction
Let f(x) = 3x² + 5x - 2 and g(x) = x² - 3x + 4
(f - g)(x) = (3x² + 5x - 2) - (x² - 3x + 4)
Distribute the negative:
= 3x² + 5x - 2 - x² + 3x - 4
Combine like terms:
= (3x² - x²) + (5x + 3x) + (-2 - 4)
= 2x² + 8x - 6
That's your answer. Factor it if you want, but leave it like this is fine.
Example 2: Rational Functions
Let f(x) = (2x + 1)/(x - 3) and g(x) = (x + 4)/(x - 3)
Since both have the same denominator, subtraction is straightforward:
(f - g)(x) = (2x + 1 - x - 4)/(x - 3)
= (x - 3)/(x - 3)
= 1 (as long as x ≠ 3)
When denominators match, just subtract the numerators. When they don't, you need a common denominator first.
Example 3: With Square Roots
Let f(x) = √x + 5 and g(x) = √x - 3
(f - g)(x) = (√x + 5) - (√x - 3)
= √x + 5 - √x + 3
= 8
The √x terms cancel out. Happens sometimes. Don't panic when it does.
Where People Go Wrong
- Forgetting to distribute the negative — This is the #1 mistake. The minus sign applies to every term in the second function.
- Dropping parentheses too early — Keep them until you've distributed the negative.
- Combining unlike terms — x² and x are different. Don't add them together.
- Ignoring domain restrictions — If g(x) has restrictions, (f - g)(x) inherits them.
Comparing Function Subtraction Methods
| Function Type | Method | Key Step |
|---|---|---|
| Polynomials | Distribute negative, combine like terms | Watch signs on every term |
| Rational Functions | Find common denominator first | Combine numerators after unifying denominators |
| Square Roots | Distribute negative, simplify radicals | Terms may cancel completely |
| Trigonometric | Subtract angle measures carefully | Use identities if needed |
| Composition | Work inside-out | Evaluate inner functions first |
Practical How-To: Subtracting Functions Step by Step
Let's do a complete problem from scratch:
Given f(x) = 4x³ - 2x² + x and g(x) = 3x³ + x² - 5x + 7, find (f - g)(x).
Step 1: Write the subtraction with parentheses
(4x³ - 2x² + x) - (3x³ + x² - 5x + 7)
Step 2: Distribute the negative to g(x)
4x³ - 2x² + x - 3x³ - x² + 5x - 7
Step 3: Group like terms
(4x³ - 3x³) + (-2x² - x²) + (x + 5x) - 7
Step 4: Combine each group
x³ - 3x² + 6x - 7
That's your answer. No need to factor unless the problem asks for it.
When to Use Function Notation
Sometimes you'll see problems written as:
Find h(x) = f(x) - g(x) if f(x) = 2x + 6 and g(x) = x - 2
This means the same thing as (f - g)(x). You can substitute directly:
h(x) = (2x + 6) - (x - 2)
h(x) = 2x + 6 - x + 2
h(x) = x + 8
Either way works. Pick whichever feels clearer for the problem.
Quick Practice Problems
Try these on your own before checking answers:
- f(x) = 5x + 3, g(x) = 2x - 4 → Find (f - g)(x)
- f(x) = x² + 4x - 1, g(x) = 3x² - x + 2 → Find (f - g)(x)
- f(x) = 6, g(x) = 2x + 1 → Find (f - g)(x)
Answers: 3x + 7, -2x² + 5x - 3, -2x + 5
The Bottom Line
Function subtraction isn't complicated. Write out both functions, distribute that negative sign, combine like terms. That's it.
The mistakes people make are always about signs and parentheses, not about the math being hard. Double-check your distribution and you'll get it right every time.