Algebra II/Number Systems & Polynomials/Operations with Functions

Algebra II Regents in 15 days

Algebra II · Lesson 4

Operations with Functions

You can add, subtract, multiply, and divide functions the same way you do with expressions.

A small business owner tracks two things every month: her revenue function $R(x)$ , which tells her total income from selling $x$ units, and her cost function $C(x)$ , which tells her total expenses. Her accountant hands her a single function — profit — and tells her it was built by subtracting one from the other. That combined function is exactly $P(x) = R(x) - C(x)$ . Operations on functions work the same way operations on numbers do: you combine the expressions and simplify.

Given two functions $f(x)$ and $g(x)$ , the four combined functions are defined as follows.

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

(f - g)(x) = f(x) - g(x)

(f \cdot g)(x) = f(x) \cdot g(x)

\left(\frac,\right)(x) = \frac,, \quad g(x) \neq 0

None of these are new operations. They are just asking you to combine two expressions you already know how to handle. The notation $(f + g)(x)$ is shorthand — it means "evaluate the sum of $f$ and $g$ at $x$ ."

Let $f(x) = 3x + 5$ and $g(x) = x^2 - 2$ . Here is how each operation plays out.

Finding (f + g)(x)

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

Expand the shorthand — write both functions added together.

= (3x + 5) + (x^2 - 2)

Replace each function with its expression. Keep parentheses until you simplify.

= x^2 + 3x + 3 \checkmark

Combine like terms: 5 and -2 give 3. Everything else stays.

Finding (f - g)(x)

(f - g)(x) = f(x) - g(x)

Same idea — now subtracting.

= (3x + 5) - (x^2 - 2)

Write both expressions. The subtraction sign in front of the second set of parentheses is about to change signs inside.

= 3x + 5 - x^2 + 2

Distribute the negative: minus times negative 2 becomes positive 2.

= -x^2 + 3x + 7 \checkmark

Combine like terms. The x² term has nothing to combine with, so it stays negative.

The most common error in function subtraction is forgetting to distribute the negative across every term in the second function. The parentheses are there to remind you that everything inside gets subtracted.

Multiplication and division follow the same structure.

Finding (f · g)(x)

(f \cdot g)(x) = f(x) \cdot g(x)

Write the product of both expressions.

= (3x + 5)(x^2 - 2)

Now it is a straightforward polynomial multiplication.

= 3x^3 - 6x + 5x^2 - 10

Distribute each term in the first factor across the second.

= 3x^3 + 5x^2 - 6x - 10 \checkmark

Reorder by degree — highest power first. No like terms here.

Finding (f/g)(x) and evaluating at x = 3

\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}

Set up the fraction.

= \frac{3x + 5}{x^2 - 2}

Plug in both expressions. This cannot be simplified further — no common factors.

\left(\frac{f}{g}\right)(3) = \frac{3(3) + 5}{(3)^2 - 2}

Substitute x = 3 into both the numerator and denominator.

= \frac{9 + 5}{9 - 2} = \frac{14}{7} = 2 \checkmark

Arithmetic in the numerator, arithmetic in the denominator, then reduce.

Division introduces one constraint the other operations do not have: the denominator cannot equal zero. For KaTeX can only parse string typed expression, any value of $x$ that makes $g(x) = 0$ must be excluded from the domain. In the example above, $g(x) = x^2 - 2 = 0$ when $x = \pm\sqrt$ , so those two values are excluded. The domain of the quotient function is all real numbers except $x = \sqrt$ and $x = -\sqrt$ .

More generally, when you build a combined function, the domain is the set of $x$ values that work in both original functions — and for division, also excludes zeros of the denominator. For most polynomial functions, this does not restrict anything. But when square roots, logarithms, or rational expressions are involved, you have to check.

Here is what all three combined functions look like on a graph, using $f(x) = 3x + 5$ and $g(x) = x^2 - 2$ .

Interactive graph — scroll to zoom, drag to pan

The blue line is $f(x)$ , the green parabola is $g(x)$ , and the red curve is $(f+g)(x)$ . Notice that for any $x$ , the height of the red curve equals the sum of the heights of the other two.

Now use $f(x) = 2x^2 + 1$ and $g(x) = x - 3$ to answer the following.

Practice Questions

(f + g)(x)

(f - g)(x)

\left(\frac{f}{g}\right)(x) \text{ and its domain}

Regents Corner

On Part II and Part III of the Algebra II Regents, operations with functions often appear alongside a request to state the domain restriction for a quotient. Writing the combined expression alone without addressing the domain earns partial credit, not full credit. Always check whether the denominator can equal zero and state the excluded value explicitly.

When computing (f - g)(x), students write f(x) - g(x) = (2x² + 1) - (x - 3) and then forget to distribute the negative, getting 2x² - x - 3 instead of 2x² - x + 4. The subtraction applies to every term inside the second set of parentheses — not just the first one.

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