Practice Function as Mapping in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Viewing a function as a mapping means thinking of it as an explicit association from each element of the domain to exactly one element of the codomain.

Like a dictionary: every word maps to a definition. Every input maps to an output.

Showing a random 20 of 50 problems.

Example 1

easy
In the dictionary mapping word→\todefinition, what plays the role of input?

Example 2

medium
A mapping diagram has an input arrow from 44 with no arrow leaving it. Is it a function on its stated domain?

Example 3

easy
Is the set of pairs {(5,8),(6,8),(7,8)}\{(5, 8), (6, 8), (7, 8)\} a function?

Example 4

medium
Restricting f(x)=x2f(x)=x^2 to x≥0x\ge 0 makes it one-to-one. What does that mean for the mapping?

Example 5

easy
Which of the following sets of ordered pairs defines a function from {1,2,3}\{1,2,3\} to R\mathbb{R}? (A) {(1,5),(2,5),(3,5)}\{(1,5),(2,5),(3,5)\} (B) {(1,2),(2,3)}\{(1,2),(2,3)\} (C) {(1,0),(2,1),(3,2),(1,4)}\{(1,0),(2,1),(3,2),(1,4)\}

Example 6

medium
Explain why the relation R={(1,2),(1,3),(2,5)}R = \{(1,2),(1,3),(2,5)\} is NOT a function from {1,2}\{1,2\} to {2,3,5}\{2,3,5\}.

Example 7

medium
Let f:R→Rf: \mathbb{R} \to \mathbb{R}, f(x)=x2f(x) = x^2. Find f−1({4})f^{-1}(\{4\}) (the pre-image of 44) and explain why ff does not have an inverse function on all of R\mathbb{R}.

Example 8

easy
Is {(1,1),(2,4),(3,9),(4,16)}\{(1, 1), (2, 4), (3, 9), (4, 16)\} a function?

Example 9

easy
Is the set of pairs {(1,2),(2,4),(3,6)}\{(1,2),(2,4),(3,6)\} a function?

Example 10

medium
A vending machine maps each button to one snack, but two buttons give chips. Function?

Example 11

medium
Domain {a,b}\{a,b\}, codomain {1,2}\{1,2\}. How many distinct functions exist?

Example 12

easy
Let f:{1,2,3}→{a,b,c}f: \{1,2,3\} \to \{a,b,c\} be defined by f(1)=af(1)=a, f(2)=af(2)=a, f(3)=cf(3)=c. Determine whether ff is a valid function, and find its range.

Example 13

medium
Let f:{1,2,3,4}→{a,b,c}f: \{1, 2, 3, 4\} \to \{a, b, c\} with f(1)=af(1) = a, f(2)=bf(2) = b, f(3)=af(3) = a, f(4)=cf(4) = c. Find the range and decide if ff is one-to-one.

Example 14

hard
Domain {1,2,3,4}\{1,2,3,4\}, codomain {a,b}\{a,b\}. How many functions are onto?

Example 15

easy
Does a vertical line graph represent a function?

Example 16

challenge
Domain {a,b,c}\{a,b,c\}, codomain {1,2}\{1,2\}. How many functions are there, and how many are one-to-one?

Example 17

medium
A mapping f:{1,2,3,4,5}→{0,1}f: \{1,2,3,4,5\} \to \{0,1\} sends nn to n mod 2n \bmod 2. List the pre-image of 11.

Example 18

hard
Define f:R→Rf: \mathbb{R} \to \mathbb{R}, f(x)=x3f(x) = x^3. Determine whether ff is one-to-one and whether it is onto.

Example 19

easy
Is {(1,2),(1,3),(2,4)}\{(1,2),(1,3),(2,4)\} a function?

Example 20

easy
Use the vertical line test: a graph hit twice by some vertical line. Function?
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