Q.Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a linear function from $\mathbb{Z}$ into $\mathbb{Z}$. Find $f(x)$.

Function Evaluation: What It Really Means

Imagine you have a machine. You drop a number into the top, the machine does something to it, and a new number comes out the bottom. That machine is a function. Function evaluation is simply the act of putting a specific input into that machine and seeing what output you get.

Let’s make it concrete. Suppose the machine’s rule is: “Take whatever number you get, multiply it by 2, then add 3.” If you drop in a 5, the machine does $2 \times 5 + 3 = 13$, and out comes 13. You have just evaluated the function at the input 5.

The Precise Statement

A function is a rule that assigns to every input (from a set called the domain) exactly one output (from a set called the codomain). We usually name a function with a letter like $f$, $g$, or $h$. The input is often written as $x$, and the output as $f(x)$ (read “f of x”).

So if $f$ is the function “multiply by 2 and add 3,” we write:

To evaluate $f$ at $x = 5$, we replace every $x$ in the expression with 5:

That’s it. Function evaluation is substitution: take the given input, plug it into the function’s rule wherever the variable appears, and simplify.

Why This Matters

Every formula you’ve ever seen — area of a circle $A(r) = \pi r^2$, distance $d(t) = vt$, even the quadratic formula — is a function. Evaluating it means you’re answering a specific question: “What happens when the input is this particular number?”

A Common Mistake

Students sometimes write $f(x+1) = 2x+3$ and then try to “solve” for $x$. Don’t. If $f(x) = 2x+3$, then $f(x+1)$ means: replace every $x$ in the rule with $(x+1)$:

The input is the whole expression $x+1$, not just $x$.

The Big Picture …