Explicit Function — Definition, Examples & Formula

Q: Can every implicit equation be converted to an explicit function?

No. Some equations cannot be algebraically solved for $y$ in terms of $x$ alone, or solving for $y$ yields multiple values for a single $x$, which violates the definition of a function. For instance, $x^2 + y^2 = 25$ gives $y = \pm\sqrt{25 - x^2}$, which is two separate explicit functions, not one.

Explicit Function

A function in which the dependent variable can be written explicitly in terms of the independent variable.

For example, the following are explicit functions: y = x² – 3, f(x) = √(x + 7) , and y = log₂ x.

Key Formula

y = f(x)

Where:

$y$ = The dependent variable, isolated on one side of the equation
$x$ = The independent variable
$f$ = A rule or expression that defines how y depends on x

Worked Example

Problem: The equation

x^2 + y = 10

defines a relationship between

x

and

y

. Write

y

as an explicit function of

x

, then evaluate

y

when

x = 3

Step 1: Isolate

y

by subtracting

x^2

from both sides of the equation.

y = 10 - x^2

Step 2: Confirm this is now in explicit form:

y

is alone on the left, and the right side contains only

x

y = f(x) = 10 - x^2

Step 3: Substitute

x = 3

into the explicit function.

y = 10 - (3)^2 = 10 - 9 = 1

Answer: The explicit function is

y = 10 - x^2

, and when

x = 3

y = 1

Another Example

This example shows that converting to explicit form sometimes requires factoring and introduces domain restrictions, unlike the simpler rearrangement in the first example.

Problem: The equation

xy - 2y = 6

defines a relationship between

x

and

y

. Convert it to an explicit function of

x

and find

y

when

x = 5

Step 1: Factor

y

out of the left side.

y(x - 2) = 6

Step 2: Divide both sides by

(x - 2)

to isolate

y

y = \frac{6}{x - 2}

Step 3: This is now explicit form. Note the restriction:

x \neq 2

because division by zero is undefined.

y = f(x) = \frac{6}{x - 2}, \quad x \neq 2

Step 4: Substitute

x = 5

y = \frac{6}{5 - 2} = \frac{6}{3} = 2

Answer: The explicit function is

y = \frac{6}{x - 2}

(with

x \neq 2

), and when

x = 5

y = 2

Frequently Asked Questions

What is the difference between an explicit function and an implicit function?

An explicit function has the dependent variable isolated on one side, like

y = 3x + 1

. An implicit function defines the relationship without isolating either variable, like

x^2 + y^2 = 25

. Some implicit equations can be rearranged into explicit form, but others (like the circle equation) cannot be written as a single explicit function because one

x

-value produces multiple

y

-values.

Can every implicit equation be converted to an explicit function?

No. Some equations cannot be algebraically solved for

y

in terms of

x

alone, or solving for

y

yields multiple values for a single

x

, which violates the definition of a function. For instance,

x^2 + y^2 = 25

gives

y = \pm\sqrt{25 - x^2}

, which is two separate explicit functions, not one.

Why are explicit functions easier to work with?

Explicit functions let you directly compute the output for any input by substituting into the formula. They are also straightforward to graph, differentiate, and integrate because the dependent variable is already isolated. Implicit equations often require special techniques like implicit differentiation.

Explicit Function vs. Implicit Function

	Explicit Function	Implicit Function
Definition	Dependent variable is isolated: $y = f(x)$	Variables are mixed together: $F(x, y) = 0$
Example	$y = x^2 - 3$	$x^2 + y^2 = 25$
Computing y	Substitute x directly into the formula	May require solving an equation for y
Differentiation	Apply standard differentiation rules to $f(x)$	Requires implicit differentiation
Always a function?	Yes, by definition	Not necessarily; may define a relation with multiple y-values per x

Why It Matters

Explicit functions appear throughout algebra, precalculus, and calculus whenever you write

y = mx + b

y = ax^2 + bx + c

, or

y = \sin x

. Recognizing whether a function is explicit helps you decide how to graph it, evaluate it, and differentiate it. In calculus courses, the distinction between explicit and implicit forms determines whether you use standard derivative rules or the more involved technique of implicit differentiation.

Common Mistakes

Mistake: Assuming every equation involving

x

and

y

can be rewritten as a single explicit function.

Correction: Some equations, like

x^2 + y^2 = 25

, produce two branches (

y = \sqrt{25 - x^2}

and

y = -\sqrt{25 - x^2}

). Each branch is an explicit function, but the original equation as a whole is not a single explicit function.

Mistake: Writing

y = f(x)

but leaving terms involving

y

on the right side.

Correction: For the function to be truly explicit, the right side must contain only the independent variable and constants — no

y

terms. If

y

still appears on both sides (e.g.,

y = x + 2y

), you have not finished isolating

y

Related Terms

Function — General concept that explicit functions are a form of
Implicit Function or Relation — Contrasting form where y is not isolated
Dependent Variable — The variable isolated in an explicit function
Independent Variable — The variable the explicit function is written in terms of
Square Root — Often arises when solving implicit equations for y
Logarithm — Example of an explicit function: y = log x