Implicit and explicit functions

1 Implicit and explicit functions

Equations such as $y = x^{2}$ , $y = \frac{1}{x}$ , $y = sin x$ are said to define $y$ explicitly as a function of $x$ because the variable $y$ appears alone on one side of the equation.

The equation

$y x + y + 1 = x$

is not of the form $y = f (x)$ but can be put into this form by simple algebra.

Task!

Write $y$ as the subject of

$y x + y + 1 = x$

We have $y (x + 1) = x - 1$ so

$y = \frac{x - 1}{x + 1}$

We say that $y$ is defined implicitly as a function of $x$ by means of $y x + y + 1 = x$ , the actual function being given explicitly as

$y = \frac{x - 1}{x + 1}$

We note than an equation relating $x$ and $y$ can implicitly define more than one function of $x$ .

For example, if we solve

$x^{2} + y^{2} = 1$

we obtain $y = \pm \sqrt{1 - x^{2}}$ so $x^{2} + y^{2} = 1$ defines implicitly two functions

$f_{1} (x) = \sqrt{1 - x^{2}} f_{2} (x) = - \sqrt{1 - x^{2}}$

Task!

Sketch the graphs of $f_{1} (x) = \sqrt{1 - x^{2}} f_{2} (x) = - \sqrt{1 - x^{2}}$

(The equation $x^{2} + y^{2} = 1$ should give you the clue.)

Since $x^{2} + y^{2} = 1$ is the well-known equation of the circle with centre at the origin and radius 1, it follows that the graphs of $f_{1} (x)$ and $f_{2} (x)$ are the upper and lower halves of this circle.

$Two graphs are displayed side-by-side, showing a function $f_1(x) = \text{positive root of } 1 - x^2$ and a function $f_2(x) = -\text{root of } 1 - x^2$.$

Long description

Sometimes it is difficult or even impossible to solve an equation in $x$ and $y$ to obtain $y$ explicitly in terms of $x$ .

Examples where explicit expressions for $y$ cannot be obtained are

$sin (x y) = y x^{2} + sin y = 2 y$

1 Implicit and explicit functions

Task!

Answer

Task!

Answer