Domain and rangeIB MYP Maths Extended: Revision notes

The domain of a function is the set of all allowed inputs ($x$-values). The range is the set of all outputs ($f(x)$ or $y$-values) that the function produces. Example: $f(x)=2x-3$ with $-1\leq x\leq4$. The domain is $-1\leq x\leq4$. Since $f(-1)=-5$ and $f(4)=5$, the range is $-5\leq f(x)\leq5$.

Use inequalities: $-1\leq x\leq4$ or $x>2$. Use set notation: $\{x\in\mathbb{R}:-1\leq x\leq4\}$, read as 'all real $x$ such that $x$ is between $-1$ and $4$ inclusive'. $\mathbb{R}$ is the set of real numbers. The symbols $\leq$ and $\geq$ include the end value; $<$ and $>$ do not. For a mapping or table, list the outputs: $1\to3,\ 2\to5,\ 3\to3$ has range $\{3,5\}$, because repeated outputs are listed once. From a graph, read how far the graph extends across ($x$) and up and down ($y$).

If no domain is given, use the largest set of real numbers that works. Two things to watch:

• Division by zero is not allowed: for $\frac{4}{x-5}$ the domain is $x\neq5$.
• Square root of a negative is not real: for $\sqrt{x+3}$ we need $x+3\geq0$, so $x\geq-3$.

A domain can be restricted by the question or by the context, for example a price that cannot be negative. Restricting the domain can change the range. For $f(x)=x^2$, the range is $f(x)\geq0$ for all real $x$, but for $-2\leq x\leq3$ the range is $0\leq f(x)\leq9$, because the greatest value occurs at $x=3$ and the least at $x=0$. To find a range on a restricted domain, work out the function at both ends, and check whether a lowest or highest point lies inside the domain.

A linear function on all real numbers has range all real numbers (unless it is constant). On a restricted domain, the range runs from the output at one end to the output at the other. A quadratic has a vertex. Complete the square to find it: $x^2-6x+5=(x-3)^2-4$, so the least value is $-4$ and the range is $f(x)\geq-4$. If the parabola opens downwards, the vertex is the greatest value. In context, write the domain and range with units and explain what they mean, for example a revenue of $0\leq R\leq800$ dirhams.