Function Notation, Domain, and Range

A function assigns exactly one output to each input. Function notation f(x) names that rule so you can evaluate it for any input, and its domain and range describe exactly which inputs are allowed and which outputs come out.

By the end you'll be able to evaluate f(x) for a given input and state a function's domain and range.

Predict: what does the machine output for f(3)? Then drag x to 3 below and check.

The machine runs \(f(x) = x^2 - 4x + 5\). Drag x and watch it get substituted into the rule step by step, producing f(x) — the same point tracks along the graph below, with dashed guides linking the input on the x-axis to the output on the curve.

x = 2.0 → (2.0)² − 4(2.0) + 5 → 4.00 − 8.00 + 5 → f(x) = 1.00
f(x) = x² − 4x + 5 — the curve, with the current input/output point
f(x) = x²−4x+5 input x output f(x)

A function is a machine that turns exactly one output out for every input you feed it — function notation f(x) gives that machine a name and lets you evaluate it, and domain and range describe what's allowed in and what actually comes out.

Intuitive

Picture f as a machine, just like the one above: you drop in a number x, the machine runs its rule, and exactly one number pops out the other side — never two, never zero. That "exactly one output" rule is what makes something a function at all. The domain is every input the machine will accept, and the range is every output it can possibly produce.

Formal

Writing \(f(x) = 2x + 1\) means "the function named f takes an input x and returns \(2x+1\)." To evaluate, substitute a number for x everywhere it appears. f(x) is a value, not multiplication — f(3) does not mean "f times 3"; it means "the output when the input is 3." Some rules make certain inputs illegal: for \(g(x)=1/(x-3)\), the input \(x=3\) forces division by zero, so the domain is every real number except 3.

Applied

Any time you name a rule and plug numbers into it — a shipping-cost calculator, a conversion formula, a model of a ball's height over time — you're using function notation. The domain tells you which inputs the model is actually meant for (you can't ship a negative number of packages), and the range tells you which outputs are even possible to get back out.

Worked example

Let \(f(x) = x^2 - 4x + 5\). Evaluate \(f(3)\) and find its domain and range. Substitute 3 for every x: \(f(3) = (3)^2 - 4(3) + 5 = 9 - 12 + 5 = \) 2. Because the rule only squares, multiplies, and adds, no input is ever illegal, so the domain is all real numbers. Completing the square gives \(f(x) = (x-2)^2 + 1\), whose smallest output is 1 (at \(x=2\)), so the range is \(f(x) \ge 1\).

Your turn

Same rule, a different input: \(f(x) = x^2 - 4x + 5\), evaluate \(f(-1)\). Substitute -1 for every x: \(f(-1) = (-1)^2 - 4(-1) + 5 = 1 + 4 + 5 = \) ____

Reveal the answer

\(f(-1) = 1 + 4 + 5 = \) 10. Set x = -1 on the slider above and check the machine's output badge and the graph point against this value.

More info — reading domain and range off the graph

On the graph above, the domain is every x-value the curve is drawn over — horizontally, it stretches forever in both directions, matching "all real numbers." The range is every y-value the curve actually reaches — vertically, it never dips below the vertex at \(y=1\), which is exactly what \(f(x)\ge 1\) means. Dragging x all the way to either end of the slider keeps the point on the curve, never below that floor. You'll bend and slide this same graph in the next lesson on shifting graphs.

Check your understanding

Question 1 of 4

Let \(f(x) = x^2 - 4x + 5\). What is \(f(0)\)?

Question 2 of 4

In function notation, what does \(f(3)\) represent?

Question 3 of 4

For \(g(x) = 1/(x - 3)\), find \(g(5)\), and state which x-value must be excluded from the domain.

Question 4 of 4

Completing the square, \(f(x) = x^2 - 4x + 5 = (x-2)^2 + 1\). What is the range of f?

Recap

  • A function assigns exactly one output to each input.
  • f(x) is a value, not multiplication: to evaluate it, substitute the input for every x in the rule and simplify.
  • The domain is the set of valid inputs; the range is the set of resulting outputs.
  • Division and square roots can shrink the domain — for \(g(x)=1/(x-3)\), \(x=3\) must be excluded.

Dive deeper

Sources

  • Function Notation, Domain, and Range