Operations with Functions

Just like numbers, functions can be added, subtracted, multiplied, and divided. When you perform an operation with two functions, \(f(x)\) and \(g(x)\), you get a brand new function whose rule is built from the outputs of the two original functions at the same input value \(x\). This idea shows up constantly in algebra 2 and precalculus, especially once you start working with polynomial, rational, and radical functions together.

Before combining functions, it helps to be comfortable evaluating a function at a specific input, whether that comes from a formula or from a table of values. Once you can evaluate \(f(x)\) and \(g(x)\) separately, combining them is just a matter of applying the operation to those two results.

Given two functions \(f(x)\) and \(g(x)\), the four arithmetic operations are defined as follows:

• 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 \(g(x) \ne 0\)

Every combined function has its own domain, and it depends on the domains of the original functions. For addition, subtraction, and multiplication, the domain of the new function is simply the set of \(x\)-values that both \(f(x)\) and \(g(x)\) share. For division, you take that same shared domain and then remove any value that makes the denominator, \(g(x)\), equal to zero.

If you need a refresher on identifying domain restrictions before combining functions, see how to write domain and range for a function.

Let \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\). Find \((f+g)(x)\) and \((f-g)(x)\).

Sum: \((f+g)(x) = (x^2 + 3x) + (2x - 1) = x^2 + 5x - 1\)

Difference: \((f-g)(x) = (x^2 + 3x) - (2x - 1) = x^2 + x + 1\)

Both \(f(x)\) and \(g(x)\) are polynomials, so their domains are all real numbers, and so is the domain of the sum and the difference.

Using the same functions, \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\), find \((f \cdot g)(x)\) and \(\left(\frac{f}{g}\right)(x)\).

Product: \((f \cdot g)(x) = (x^2 + 3x)(2x - 1) = 2x^3 + 5x^2 - 3x\)

Quotient: \(\left(\frac{f}{g}\right)(x) = \dfrac{x^2 + 3x}{2x - 1}\), which is undefined when \(2x - 1 = 0\), that is, when \(x = \frac{1}{2}\). So the domain of the quotient excludes \(x = \frac{1}{2}\).

Consider \(f(x) = x^2\) and \(g(x) = x + 1\). Their sum, \((f+g)(x) = x^2 + x + 1\), is a new function you can graph on its own. At \(x = 2\), \((f+g)(2) = f(2) + g(2) = 4 + 3 = 7\), which matches the point marked on the graph below.

It is easy to confuse combining functions with composing them. The operations covered here, addition, subtraction, multiplication, and division, combine the outputs of two functions at the same input. Composition of functions is a different process where the output of one function becomes the input of another, written \(f(g(x))\). Related applications, such as building a difference quotient from a function and itself, use these same operation skills; see difference quotient: applications of functions for that next step.

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