How to write a relation as a function in f(x) notation

Key concept: This kind of question is really about reading the equation as a rule. Once you isolate $y$, you can usually write the function directly.

Step by step

To rewrite a relation using $f:X\to Y$ notation, identify the input, the output, and the formula.

a. $\{(x,y)\mid y=3x+2\}$

This is a function from real numbers to real numbers:

$f:\mathbb{R}\to\mathbb{R},\quad f(x)=3x+2$

b. $\{(x,y)\mid 2y+3x=12\}$

Solve for $y$:

$2y=12-3x$ $y=6-\frac{3}{2}x$

So

$f:\mathbb{R}\to\mathbb{R},\quad f(x)=6-\frac{3}{2}x$

c. $\{(x,y)\mid y=2x+3,\ x\ge 0\}$

The formula is the same, but the domain is restricted:

$f:[0,\infty)\to\mathbb{R},\quad f(x)=2x+3$

The key idea is that the function rule and the domain both matter.

Pitfall alert

Don’t forget the domain restriction in part c. If $x\ge 0$, then the function is not from all real numbers. Also, for part b, leaving the equation as $2y+3x=12$ is not enough—you need $y$ alone for standard function notation.

Try different conditions

If the relation cannot be solved for a single output value, then it is not a function. For example, $x=y^2$ gives two possible $y$-values for many $x$-values, so it fails the function test unless the domain is restricted further.

Further reading

function notation, domain and range, relation