Function domains, inverses and graph transformations

Follow a number through a rule

Think of a function (funzione) as a rule with a specified set of allowed inputs, its domain (dominio). Each allowed input produces exactly one output. Different inputs may share an output. The range (immagine) consists of the outputs that actually occur. For example, squaring every real number produces every nonnegative real number and never produces a negative one. Restrict the inputs to numbers between two and three, and the same formula now has outputs between four and nine, including the endpoints if the input interval includes them.

When finding a natural real domain, read an expression from the innermost operation outwards. The argument of a square root must be nonnegative. A denominator must differ from zero. These conditions combine if a square root appears below a fraction bar, since the root must exist and also have a nonzero value. Cancellation may give a shorter expression with the same values on its original domain, but the simplified formula alone can conceal an excluded input. Write restrictions beside the formula before simplifying it.

Compose and reverse

In a composition (composizione) the output of the inner rule becomes the input to the outer one. For $f(g(x))$ you need permission to use $x$ in $g$ and permission to use $g(x)$ in $f$. Suppose the inner operation subtracts five and the outer operation takes a square root. You can start with seven because the intermediate value is two. Starting with four fails at the second operation. Reversing the operations puts the initial restriction on the original input instead and gives a different final result.

An inverse function (funzione inversa) retraces the journey from an output to its unique input. You can draw a horizontal line at the proposed output value to count the inputs that produced it. If it crosses at two points, reversing the rule on that domain would give two answers for a single input. Restricting the domain to an increasing or decreasing branch can fix the problem. For the square function restricted to nonnegative inputs the inverse takes the positive square root or zero. Restrict it to nonpositive inputs instead and the inverse takes the negative of that root. Its domain in either case is the original range.

Move an individual point

Suppose the original graph contains the point $(a,b)$. This records $f(a)=b$. In $y=f(x-h)+k$ the inner expression must equal $a$ to recover the same old output. Solve $x-h=a$ to find the new horizontal coordinate $a+h$, then add $k$ to the output. The point moves to $(a+h,b+k)$. This argument works for an endpoint or a peak just as well as for an ordinary point. Following one recognisable point is often easier than redrawing a complete graph.

Absolute values give two different reflections depending on their position. With $y=|f(x)|$, every input stays put. A negative height becomes positive and the part already above the horizontal axis stays in place. With $y=f(|x|)$ both positive and negative inputs ask for the old output at a nonnegative coordinate. Keep the original right-hand portion and reflect it across the vertical axis. Where the right-hand input was unavailable there will be no corresponding point on either side.

Test symmetry on the actual domain

An even function (pari) satisfies $f(-x)=f(x)$ and reflects into itself across the vertical axis. An odd function (dispari) satisfies $f(-x)=-f(x)$ and its graph survives a half-turn about the origin. Both definitions require the domain to contain opposite pairs of inputs. A square restricted to positive inputs therefore lacks the full symmetry of the ordinary square function on the real line. One mismatching pair disproves a claimed symmetry. Matching values at just one pair are a clue for a sketch, but a proof must cover every input in the domain.