Subtracting Functions and the Negative Trap
Subtracting functions in Algebra 1 requires careful distribution of the negative sign across every term — a common error point in California Reveal Math, Algebra 1 (Grade 9). When computing (f-g)(x), always place g(x) inside parentheses: f(x) - (g(x)). Then distribute the negative sign to flip every sign in g(x) before combining like terms. For example, if f(x) = 3x²+5x+1 and g(x) = x²-4x+6, then (f-g)(x) = 3x²+5x+1-(x²-4x+6) = 3x²+5x+1-x²+4x-6 = 2x²+9x-5. The classic mistake is only flipping the sign of the first term of g. This skill builds accuracy in polynomial operations essential for factoring and solving quadratics.
Key Concepts
Property When finding the difference of two functions, $(f g)(x)$, you must substitute the entire expression for $g(x)$ inside parentheses. The negative sign must then be distributed across every single term of $g(x)$ before combining like terms.
Examples Distributing the Negative: Let $f(x) = 3x^2 + 5x + 1$ and $g(x) = x^2 4x + 6$. Find $(f g)(x)$. Setup with parentheses: $3x^2 + 5x + 1 (x^2 4x + 6)$ Distribute the negative (flip all signs in $g$): $3x^2 + 5x + 1 x^2 + 4x 6$ Combine like terms: $2x^2 + 9x 5$ The Common Error: A student writes $3x^2 + 5x + 1 x^2 4x + 6$. This is incorrect. The student only subtracted the first term of $g(x)$ and forgot to flip the signs of $ 4x$ and $6$.
Common Questions
How do you subtract two functions to find (f-g)(x)?
Write f(x) - (g(x)) with g(x) in parentheses, then distribute the negative sign to flip every sign in g(x), and finally combine like terms.
What is the most common error when subtracting functions?
Only subtracting the first term of g(x) and forgetting to flip the signs of all remaining terms. Always put g(x) in parentheses first.
Find (f-g)(x) if f(x)=3x²+5x+1 and g(x)=x²-4x+6.
(f-g)(x) = 3x²+5x+1-(x²-4x+6) = 3x²+5x+1-x²+4x-6 = 2x²+9x-5.
Why must you use parentheses when writing g(x) in a function subtraction?
The minus sign applies to the entire function g(x), not just its first term. Parentheses signal that every term inside must have its sign flipped when the negative is distributed.
How is subtracting functions different from adding functions?
Addition (f+g)(x) = f(x)+g(x) combines like terms directly. Subtraction requires distributing a negative sign through all of g(x) first, which flips the signs of all its terms.
What is the domain of (f-g)(x)?
The domain is the intersection of the domains of f and g. For polynomials, this is all real numbers.
How does function subtraction connect to polynomial operations?
(f-g)(x) is equivalent to subtracting two polynomials: distribute the negative sign across g's polynomial, then combine like terms — the same skill used throughout algebra.