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If $f(x)$ is a function defined only for $0 \le x \le 1$, and $f(x) = ax+b$ for constants $a$ and $b$ where $a < 0$, then what is the range of $f$ in terms of $a$ and $b$? Express your answer in interval notation.

michaelcai  Aug 28, 2017

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 #1
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f(x)  is defined only for  0 ≤ x ≤ 1  , and  f(x)  =  ax + b  for constants  a  and  b  where  a < 0  .

 

 

Since  a  is negative, the smallest possible value for  f(x)  will be when  x = 1  .

f(1)  =  a(1) + b  =  a + b

 

Then, the largest possible value for  f(x)  will be when  x = 0  .

f(0)  =  a(0) + b  =  b

 

 

The smallest possible value for  f(x)  is  a + b  , and the largest possible value for  f(x)  is  b  .

 

So......the range for  f(x)  is  [ a + b ,  b ]  .

hectictar  Aug 29, 2017
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 #1
avatar+4470 
+4
Best Answer

f(x)  is defined only for  0 ≤ x ≤ 1  , and  f(x)  =  ax + b  for constants  a  and  b  where  a < 0  .

 

 

Since  a  is negative, the smallest possible value for  f(x)  will be when  x = 1  .

f(1)  =  a(1) + b  =  a + b

 

Then, the largest possible value for  f(x)  will be when  x = 0  .

f(0)  =  a(0) + b  =  b

 

 

The smallest possible value for  f(x)  is  a + b  , and the largest possible value for  f(x)  is  b  .

 

So......the range for  f(x)  is  [ a + b ,  b ]  .

hectictar  Aug 29, 2017

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