If we know range of a function , how can construct range of other function?If we know − 1 2 ≤ x x 2 + 1 ≤ 1 2 how can obtain the range of the function below? y = 4 x 9 x 2 + 25 The problem is what's the range with respect to − 1 2 ≤ x x 2 + 1 ≤ 1 2

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If we know range of a function , how can construct range of other function?If we know             −      1        2    ≤      x                  x        2            +      1        ≤      1    2   how can obtain the range of the function below?  y  =            4      x              9              x        2            +      25      The problem is what&#039;s the range with respect to             −      1        2    ≤      x                  x        2            +      1        ≤      1    2

Peyton Atkins · Accepted Answer

Write            4      x              9              x        2            +      25        =      4          9      x      +              25        x            and use AM-GM in two cases:1)   x  &amp;gt;  0  y  ≤      4          2              9        x        ⋅                  25          x                      =      2    15    .2)   x  &amp;lt;  0:  y  ≥  −      4          2              (        −        9        x        )        ⋅                  (          −                      25            x                    )                      =  −      2    15    .The equality occurs for   9  x  =      25    x   and since we have a continuous function,we got the answer:      [    −          2      15        ,          2      15        ]

acapotivigl · Accepted Answer

y  =            4      x              9              x        2            +      25          y  =            4      x              25      (              9        25                    x        2            +      1      )        =    y  =      4    25        x          (      (              3        5            x              )        2            +      1      )      now take   u  =            3      x        5    y  =      4    25        x          (      (              3        5            x              )        2            +      1      )        =        4    25                      5        3            u              (      (      u              )        2            +      1      )        =        4    15        u                  u        2            +      1        you know             −      1        2    ≤      u                  u        2            +      1        ≤      1    2   so  −      4    15    ×      1    2    ≤  y  =      4    15        u                  u        2            +      1        ≤      4    15    ×      1    2

If we know range of a function , how can construct range of other function? If we know (-1)/(2)<= (x)/(x^2+1)<= (1)/(2) how can obtain the range of the function below? y=(4 x)/(9x^2+25)

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