Functional Dependency Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Functional Dependency.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A relationship where the value of one quantity (the output or dependent variable) is completely determined by the value of another quantity (the input or independent variable). If $y$ depends on $x$ , then knowing $x$ uniquely determines $y$ .

Temperature determines ice cream sales—sales DEPEND ON temperature.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Functional dependency means fixing the input forces a single determined output.

Common stuck point: The procedure for functional dependency is the easy part; the trap is allowing one input to give two outputs and still calling it a function. Asking "Does every allowed input produce exactly one output (never two)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does every allowed input produce exactly one output (never two)?

Worked Examples

Example 1

easy

y

functionally dependent on

x

in the equation

y = 3x + 1

Answer

Yes,

y

depends functionally on

x

First step

Step 1: For any

x

, there is exactly one

y = 3x + 1

Full solution

2
Step 2: $x = 0 \to y = 1$ , $x = 1 \to y = 4$ . Each input gives one output.
3
Step 3: Yes, $y$ is a function of $x$ .

Functional dependency means each input determines exactly one output. The equation

y = 3x + 1

passes the vertical line test — every

x

maps to a unique

y

Example 2

medium

y

functionally dependent on

x

x^2 + y^2 = 25

Example 3

medium

y

depends on

x

via

y = 2^x

. If

x = 3

, what is

y

, and is

x

uniquely determined by

y

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Does

\{(1,2), (1,3), (2,4)\}

represent a function?

Example 2

medium

In a class, is final grade a function of student name?

Example 3

easy

y = 3x

and

x = 4

, what is

y

? Does

x

determine

y

Example 4

easy

\{(1,2),(2,4),(3,6)\}

, is

y

a function of

x

Example 5

easy

\{(1,2),(1,5),(3,6)\}

, is

y

a function of

x

Example 6

easy

Which is the independent variable in 'ice cream sales depend on temperature'?

Example 7

easy

f(x)=x^2

, what is

f(3)

Example 8

easy

In the equation

x + y = 10

, if

x = 4

, what is

y

? Does

x

determine

y

here?

Example 9

easy

Does

y

depend on

x

y = 7

(a constant function)?

Example 10

easy

Correlation vs dependency: shoe size and reading ability both rise with age in children. Does shoe size functionally determine reading ability?

Example 11

medium

Given

f(x) = 2x + 1

, find the input

x

for which

f(x) = 11

Example 12

medium

For

f(x)=\sqrt{x-3}

, what inputs

x

are allowed (the domain)?

Example 13

medium

Two variables satisfy

y = x^2

. If

y = 9

, is

x

uniquely determined?

Example 14

medium

In the relation

x = y^2

, is

y

a function of

x

? Use the vertical line test idea.

Example 15

medium

A table gives

f

f(0)=1, f(1)=1, f(2)=1

. Is

f

a valid function even though outputs repeat?

Example 16

medium

z

depends on

y

and

y

depends on

x

via

y=x+1

z=y^2

, express

z

as a function of

x

and find

z

when

x=2

Example 17

medium

Does the equation

x^2 + y^2 = 25

define

y

as a function of

x

Example 18

medium

Given

f(x) = \frac{1}{x-2}

, identify the input that is NOT allowed and explain in terms of dependency.

Example 19

medium

Given

g(x) = 5 - 2x

, find the input that maps to output 5, and the output when input is 5.

Example 20

challenge

Determine all values of

k

for which

y = kx^2 + (k-1)x + 1

makes

y

depend on

x

as a genuine quadratic (not linear/constant), and find

y(1)

in terms of

k

Example 21

challenge

For

f(x)=\frac{ax+b}{cx+d}

to be its own inverse (

f(f(x))=x

), what relation must

a

and

d

satisfy? Give one nontrivial condition.

Example 22

challenge

A relation

R

\{1,2,3\}

\{(1,2),(2,3),(3,1)\}

. Does

R

define

y

as a function of

x

, and is the resulting function invertible?

Example 23

easy

\{(2,5),(3,5),(4,5)\}

a function?

Example 24

easy

\{(0,1),(0,-1)\}

a function?

Example 25

easy

In the relation 'student ID determines exam score', which is the dependent variable?

Example 26

easy

Does the vertical line

x = 5

define

y

as a function of

x

Example 27

easy

True or false: every function is a relation.

Example 28

easy

h(x) = 5

for every real

x

, is

h

a function?

Example 29

medium

For

f(x) = \sqrt{4 - x}

, find the domain.

Example 30

medium

For

g(x) = \frac{1}{x^2 - 9}

, find the domain.

Example 31

medium

Does

y = \pm\sqrt{x}

make

y

a function of

x

Example 32

medium

A table lists 30 students by name with their birth month. Is birth month a function of name?

Example 33

medium

Is birth month a function of birth date (year-month-day)?

Example 34

medium

y

a function of

x

given by the graph

y = |x|

Example 35

medium

x + y = 7

, is

y

a function of

x

Example 36

medium

Does the graph of a circle

x^2 + y^2 = 4

pass the vertical line test?

Example 37

hard

For

f(x) = \frac{x^2-1}{x-1}

, find the domain and simplify where defined.

Example 38

hard

Does

y^2 = x

define

y

as a function of

x

? Does it define

x

as a function of

y

Example 39

hard

For

f(x) = \frac{\sqrt{x+1}}{x-2}

, state the domain.

Example 40

hard

A weather model: temperature depends on time of day. Suppose at 3 PM the temperature is recorded as both 71°F and 73°F by two thermometers. Is temperature a function of time here?

Example 41

hard

z = f(x, y) = x + y

and

y

is held constant at

3

, is

z

a function of

x

alone?

Example 42

challenge

For what real

a

does

y = ax^2 + (1-a)x + a

define a one-to-one function on all reals?

Example 43

challenge

Database analogue: a table has columns (student_id, name). If student_id is the primary key, is name functionally dependent on student_id?

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Related Concepts

Function Variables Function Families Causation

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definitionvariables