Finding the domain

Task

For the function $$f(x) = \frac{2}{x-3}$$

1. Evaluate f(11), writing out every step. Write the output in decimal form.
2. Evaluate f(3), writing out every step. You will run into some trouble—describe it.
3. When you evaluate this function at an input, what operations are performed, and in what order? List them. What restrictions does each operation place on the domain of the function?
4. Give a possible domain for $f$.

IM Commentary

The purpose of this task is to introduce the idea of the domain of a function by linking it to the evaluation of an expression defining the function. By thinking through the evaluation step by step, students isolate the exact point where a given input results in an undefined output. In part (b), any domain that excludes $x=3$ is possible. It is conventional when given a function defined by an expression to take the domain to be the largest possible, but it is worth pointing that this is a convention, not a mathematical fact. As students gain a mature understanding of functions they learn that the domain is something that is specified when you define the function, it does not come already attached.

The function in this task can be broken down into two simple operations. More complicated variations are possible.

This task is adapted from Algebra: Form and Function, McCallum et al., Wiley 2010.

Solution

1. $$f(x)=\frac{2}{x-3}$$ $$f(11)=\frac{2}{11-3}$$ $$f(11)=\frac{2}{8}$$ $$f(11)=0.25$$ When the input is $11$, the output is $0.25$. Said another way, $11$ is an element of the domain, and $0.25$ is an element of the range.
2. $$f(x)=\frac{2}{x-3}$$ $$f(3)=\frac{2}{3-3}$$ $$f(3)=\frac{2}{0}$$ The trouble is that $\frac{2}{0}$ is an undefined value. So, $3$ is not an element of the domain.
3. For an input $x$ you calculate the output $f(x)$ in the following steps: subtract 3 from $x$, then divide the result, $x-3$, into 2. The first operation is always valid, but the second is only valid if $x -3 \neq 0$, so the domain cannot include $x = 3$.
4. One possible answer is all real numbers except $x = 3$. Any subset of this domain is also possible, e.g., all negative numbers, or all positive numbers except $x=3$.