A function f has domain [0, 2] and range [0, 1]. (The notation [a, b] denotes {x : a ≤ x ≤ b}. ) What are the domain and range, respectively, of the function g defined by g(x) = 1 - f(x + 1)?

2008 AMC 12A Problem 12

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12.

A function

f

has domain

[0, 2]

and range

[0, 1].

(The notation

[a, b]

denotes

\{x : a \le x \le b\}.

) What are the domain and range, respectively, of the function

g

defined by

g(x) = 1 - f(x + 1)?

[-1, 1],

[-1, 0]

[-1, 1],

[0, 1]

[0, 2],

[-1, 0]

[1, 3],

[-1, 0]

[1, 3],

[0, 1]

Concepts:function

f(x + 1)

is defined when

0 \le x + 1 \le 2

Subtracting

f(x + 1) \in [0, 1]

from

1

reverses the interval but keeps it

[0, 1]

The value

f(x + 1)

is defined when

0 \le x + 1 \le 2,

that is,

-1 \le x \le 1,

so the domain of

g

[-1, 1].

f(x + 1)

ranges over

[0, 1],

the value

1 - f(x + 1)

ranges over

[0, 1]

as well, so the range of

g

[0, 1].

Thus, B is the correct answer.