SOLUTION: 1.Analytically show that the function is one-to-one, find its inverse, and evaluate the following: f(x)=10-∛(x-8) f^(-1) (10) f^(-1) (11) f^(-1) (12)

Algebra ->  Inverses -> SOLUTION: 1.Analytically show that the function is one-to-one, find its inverse, and evaluate the following: f(x)=10-∛(x-8) f^(-1) (10) f^(-1) (11) f^(-1) (12)      Log On


   

Question 1125641: 1.Analytically show that the function is one-to-one, find its inverse, and evaluate the following:
f(x)=10-∛(x-8)

f^(-1) (10)
f^(-1) (11)
f^(-1) (12)
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
find its inverse, and evaluate the following:
f%28x%29=10-root%283%2Cx-8%29
first
f%28x%29=y
y=10-root%283%2Cx-8%29....Interchange the variables:

x=10-root%283%2Cy-8%29...solve for y
root%283%2Cy-8%29=10-x
%28root%283%2Cy-8%29%29%5E3=%2810-x%29%5E3
y-8=-x%5E3+%2B+30+x%5E2+-+300+x+%2B+1000
y=-x%5E3+%2B+30+x%5E2+-+300+x+%2B+1000%2B8 , so your inverse is
f%5E%28-1%29=-x%5E3+%2B+30+x%5E2+-+300+x+%2B+1008

f%5E%28-1%29+%2810%29=-%2810%29%5E3+%2B+30+%2810%29%5E2+-+300+%2810%29+%2B+1008
f%5E%28-1%29+%2810%29=-%281000%29+%2B+30+%28100%29+-+300+%2810%29+%2B+1008
f%5E%28-1%29+%2810%29=-1000+%2B+3000++-3000++%2B+1008
f%5E%28-1%29+%2810%29=-1000++%2B+1008
f%5E%28-1%29+%2810%29=8


f^(-1) (11)
f%5E%28-1%29+%2811%29=-%2811%29%5E3+%2B+30+%2811%29%5E2+-+300+%2811%29+%2B+1008
f%5E%28-1%29+%2811%29=-%281331%29+%2B+30+%28121%29+-+300+%2811%29+%2B+1008
f%5E%28-1%29+%2811%29=-1331+%2B+3630++-3300++%2B+1008
f%5E%28-1%29+%2811%29=-4631++%2B+4638
f%5E%28-1%29+%2811%29=7




f%5E%28-1%29+%2812%29=-%2812%29%5E3+%2B+30+%2812%29%5E2+-+300+%2812%29+%2B+1008
f%5E%28-1%29+%2812%29=-1728+%2B+4320+-+3600+%2B+1008
f%5E%28-1%29+%2812%29=+5328+-+5328+
f%5E%28-1%29+%2812%29=+0+