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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\$

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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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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
+4470
+4

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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