1) Find f − 1 ( x ), if possible.2) Factor the polynomial and use the factored form to find the zeros. Also describe the end behavior. f ( x ) = − 4 x 3 + 14 x 2 + 30 x

Question

1) Find       f          −      1        (  x  ), if possible.2) Factor the polynomial and use the factored form to find the zeros. Also describe the end behavior.  f  (  x  )  =  −  4      x    3    +  14      x    2    +  30  x

Jazlene Dickson · Accepted Answer

Step 11)       f          −      1        (  x  )  y  =  f  (  x  )  ⇒  y  =  f  (  x  )  ⇒  y  =  3  −  (  x  +  4      )    2  Interchange   x  →  y     and     y  →  x  ⇒  x  =  3  −  (  y  +  4      )    2    x  =  3  −  (  y  +  4      )    2    ⇒  x  −  3  =  −  (  y  +  4      )    2      ⇒  −  (  3  −  x  )  =  −  (  y  +  4      )    2      ⇒  (  3  −  x  )  =  (  y  +  4      )    2      ⇒      3    −    x    =  y  +  4    ⇒      3    −    x    −  4  =  y    ⇒  y  =      3    −    x    −  4Step 2Since   x  ≤  −  4Thus       F          −      1        (  x  ) will existand       f          −      1        (  x  )  =      3    −    x    −  4

Donna Flynn · Accepted Answer

Step 12)   f  (  x  )  =  −  4      x    3    +  14      x    2    +  30  x    =  −  2  x  (  2      x    2    −  7  x  −  15  )    =  −  2  x  (  2      x    2    −  10  x  +  3  x  −  15  )    =  −  2  x  [  2  x  (  x  −  5  )  +  3  (  x  −  5  )  ]    =  −  2  x  (  (  x  −  5  )  (  2  x  +  3  )  )    f  (  x  )  =  −  2  x  (  x  −  5  )  (  2  x  +  3  )Step 2Thus   f  (  x  )  =  −  4      x    3    +  14      x    2    +  30  x  =  −  2  x  (  x  −  5  )  (  2  x  +  3  )Now zeros  f  (  x  )  =  0    −  2  x  (  x  −  5  )  (  2  x  +  3  )  =  0    −  2  x  =  0  ,  x  −  5  =  0  ,  2  x  +  3  =  0    x  =  0  ,  x  =  5  ,  x  =            −      3        2      Z  e  r  o  s  =  0  ,  5  ,            −      3        2  As   x  →  −  ∞  ,  y  →  +  ∞As   x  →  +  ∞  ,  y  →  −  ∞

Find f^{-1}(x), if possible.

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