Argument of a Function

Argument of a Function

The variable, term or expression on which a function operates. For example, the argument of is x, the argument of sin(2A) is 2A, and the argument of e^{x – 5} is x – 5. The argument of f(x) is x.

See also

Sine, e

Example

Problem: Identify the argument of each function: (a) f(x) = x² + 1, evaluated at x = 3; (b) ln(5t − 2); (c) cos(π/4).

Part (a): The function f takes an input and squares it, then adds 1. When we write f(3), the argument is the value we substitute for x.

f(3) = 3^2 + 1 = 10 \quad \Rightarrow \quad \text{argument} = 3

Part (b): The natural logarithm function ln operates on whatever appears inside the parentheses. Here the entire expression 5t − 2 is being fed into ln.

\ln(5t - 2) \quad \Rightarrow \quad \text{argument} = 5t - 2

Part (c): The cosine function acts on the quantity inside the parentheses. Even though π/4 is a constant, it is still the argument.

\cos\!\left(\frac{\pi}{4}\right) \quad \Rightarrow \quad \text{argument} = \frac{\pi}{4}

Answer: The arguments are 3, 5t − 2, and π/4, respectively. The argument is always the complete expression sitting inside the function's parentheses.

Another Example

Problem: Given g(x) = e^(x² − 9), identify the argument of the exponential function and evaluate g(3).

Step 1: The base-e exponential function acts on whatever is in its exponent. That entire exponent is the argument.

e^{x^2 - 9} \quad \Rightarrow \quad \text{argument} = x^2 - 9

Step 2: Substitute x = 3 into the argument to get its specific value.

x^2 - 9 = 3^2 - 9 = 0

Step 3: Evaluate the function with that argument value.

g(3) = e^{0} = 1

Answer: The argument of the exponential function is x² − 9. When x = 3, the argument equals 0, so g(3) = 1.

Frequently Asked Questions

What is the difference between an argument and a parameter of a function?

An argument is the specific value or expression you pass into a function when you use it. A parameter is the placeholder variable used when you define the function. In f(x) = 2x + 1, x is the parameter; when you compute f(5), the number 5 is the argument. In everyday math classes, the two words are often used interchangeably, but the distinction matters in formal contexts and programming.

Can a function have more than one argument?

Yes. A function of two or more variables has multiple arguments. For example, h(x, y) = x² + y² has two arguments: x and y. When you evaluate h(3, 4), the arguments are 3 and 4.

Argument vs. Variable

A variable is a symbol that can take different values, such as x. An argument is the specific input — which may be a variable, a number, or a whole expression — that you place inside a function. In f(x), the variable x serves as the argument. In f(7), the constant 7 is the argument. So every variable used as a function input is an argument, but not every argument is a single variable.

Why It Matters

Correctly identifying the argument is essential when applying rules like the chain rule in calculus, where you differentiate the outer function and then multiply by the derivative of its argument. It also matters when solving equations: to "undo" a function, you isolate its argument first. In programming and science, the term appears constantly — every time you call a function, you supply its arguments.

Common Mistakes

Mistake: Identifying only part of the argument instead of the whole expression inside the parentheses.

Correction: The argument is everything the function operates on. In sin(3x + π), the argument is the entire expression 3x + π, not just 3x or x alone.

Mistake: Confusing the argument with the output (value) of the function.

Correction: The argument is the input. In f(4) = 10, the argument is 4 and the output is 10. These are different roles.

Related Terms

Function — A rule that acts on its argument
Variable — Often serves as the argument
Expression — An argument can be any expression
Term — A single-part expression usable as argument
Domain — Set of all allowable arguments
Sine — Common trig function with one argument
e — Base of exponential function taking an argument