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Difference between revisions of "2008 AMC 12A Problems/Problem 6"

(New page: ==Problem == Consider a function <math>f(x)</math> with domain <math>[0,2]</math> and range <math>[0,1]</math>. Let <math>g(x)=1-f(x+1)</math>. What are the domain and range, respectively,...)
 
(correct problem)
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==Problem ==
 
==Problem ==
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Consider a function <math>f(x)</math> with domain <math>[0,2]</math> and range <math>[0,1]</math>. Let <math>g(x)=1-f(x+1)</math>. What are the domain and range, respectively, of <math>g(x)</math>?
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Heather compares the price of a new computer at two different stores. Store A offers <math>15\%</math> off the sticker price followed by a <dollar/><math>90</math> rebate, and store B offers <math>25\%</math> off the same sticker price with no rebate. Heather saves <dollar/><math>15</math> by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?
  
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<math>\textbf{(A)}\ [ - 1,1],[ - 1,0] \qquad \textbf{(B)}\ [ - 1,1],[0,1] \qquad \textbf{(C)}\ [0,2],[ - 1,0] \qquad \textbf{(D)}\ [1,3],[ - 1,0] \qquad \textbf{(E)}\ [1,3],[0,1]</math>
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<math>\textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500</math>
  
 
==Solution==
 
==Solution==
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<math>g(x)</math> is defined iff <math>f(x + 1)</math> is defined. Thus the domain is all <math>x| x + 1 \in [0,2] \rightarrow x \in [ - 1,1]</math>.  
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Let <math>S</math> be the sticker price.  
  
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Since <math>f(x + 1) \in [0,1]</math>, <math>- f(x + 1) \in [ - 1,0]</math>. Thus <math>g(x) = 1 - f(x + 1) \in [0,1]</math> is the range of <math>g(x)</math>.  
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Store A sells the computer for <math>A=S-0.15S-90=0.85S-90</math> dollars.  
 +
Store B sells the computer for <math>B=S-0.25S=0.75S</math> dollars.  
  
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Thus the answer is <math>[ - 1,1],[0,1] \Rightarrow B</math>.  
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Since she saves <dollar/><math>15</math> at store A, <math>15=B-A=0.75S-(0.85S-90)=-0.1S+90</math>. Thus, <math>S=750 \Rightarrow A</math>.  
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2008|ab=A|num-b=5|num-a=7}}
 
{{AMC12 box|year=2008|ab=A|num-b=5|num-a=7}}

Revision as of 20:33, 18 February 2008

Problem

Heather compares the price of a new computer at two different stores. Store A offers $15\%$ off the sticker price followed by a <dollar/>$90$ rebate, and store B offers $25\%$ off the same sticker price with no rebate. Heather saves <dollar/>$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?

$\textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$

Solution

Let $S$ be the sticker price.

Store A sells the computer for $A=S-0.15S-90=0.85S-90$ dollars. Store B sells the computer for $B=S-0.25S=0.75S$ dollars.

Since she saves <dollar/>$15$ at store A, $15=B-A=0.75S-(0.85S-90)=-0.1S+90$. Thus, $S=750 \Rightarrow A$.

See Also

2008 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions
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