1 Implicit and explicit functions

Equations such as y = x 2 , y = 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 = 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 = 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 = ± 1 x 2 so x 2 + y 2 = 1 defines implicitly two functions

f 1 ( x ) = 1 x 2 f 2 ( x ) = 1 x 2

Task!

Sketch the graphs of f 1 ( x ) = 1 x 2 f 2 ( x ) = 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.

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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