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 — Solution

2008 AMC 12A Problem 12

Below is the professionally curated solution for Problem 12 of the 2008 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12A solutions, or check the answer key.

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Concepts:function

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

$[-1, 1], [-1, 0]$

$[-1, 1], [0, 1]$

$[0, 2], [-1, 0]$

$[1, 3], [-1, 0]$

$[1, 3], [0, 1]$

Solution:

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 $g$ is $[-1, 1].$

As $f(x + 1)$ ranges over $[0, 1],$ the value $1 - f(x + 1)$ ranges over $[0, 1]$ as well, so the range of $g$ is $[0, 1].$

Thus, B is the correct answer.

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