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

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


   

Question 1125606: Analytically show that the function f(x)=10-∛(x-8)
is one-to-one, find its inverse, and evaluate the following:
f^(-1) (10)
f^(-1) (11)
f^(-1) (12)



Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

If f(x) is one-to-one, then f%28a%29=f%28b%29 implies that a+=+b for all a,b in the domain of f(x). Put in plain english: if two output values are the same, then the inputs must be the same for the function to be one-to-one.

Consider a counter-example such as a parabola. We can have the same output lead to two different inputs (eg: y = 4 lead to x = -2 and x = 2 for the function y = x^2). This is a reason why a parabola is not one-to-one.

f%28x%29+=+10-root%283%2Cx-8%29

f%28a%29+=+10-root%283%2Ca-8%29

f%28b%29+=+10-root%283%2Cb-8%29

--------------------------------------------

If f%28a%29+=+f%28b%29, then...

f%28a%29+=+f%28b%29

10-root%283%2Ca-8%29+=+10-root%283%2Cb-8%29 Substitution

-root%283%2Ca-8%29+=+-root%283%2Cb-8%29 Subtract 10 from both sides

root%283%2Ca-8%29+=+root%283%2Cb-8%29 Multiply both sides by -1

a-8+=+b-8 Cube both sides

a+=+b Add 8 to each side

So we end up with a+=+b after assuming f%28a%29+=+f%28b%29

So this means that f%28a%29+=+f%28b%29 leads to a+=+b. If we follow the steps shown above in reverse, then we'll go from a+=+b to f%28a%29+=+f%28b%29

Therefore, we have proven that f(x) is indeed one-to-one.

--------------------------------------------

Let's find the inverse. Which I'll call g(x)

g(x) = f^(-1)(x)

f%28x%29+=+10-root%283%2Cx-8%29

y+=+10-root%283%2Cx-8%29 Replace f(x) with y

x+=+10-root%283%2Cy-8%29 Swap x and y. From here on out, we're solving for y.

x-10+=+-root%283%2Cy-8%29 Subtract 10 from both sides

-root%283%2Cy-8%29+=+x-10

root%283%2Cy-8%29+=+-x%2B10 Multiply both sides by -1

y-8+=+%28-x%2B10%29%5E3 Cube both sides

y+=+%28-x%2B10%29%5E3%2B8 Add 8 to both sides

The inverse is g%28x%29+=+%28-x%2B10%29%5E3%2B8

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Let's use the inverse to compute the inputs x = 10, x = 11, x = 12

g%28x%29+=+%28-x%2B10%29%5E3%2B8

g%2810%29+=+%28-10%2B10%29%5E3%2B8

g%2810%29+=+%280%29%5E3%2B8

g%2810%29+=+0%2B8

g%2810%29+=+8

------

g%28x%29+=+%28-x%2B10%29%5E3%2B8

g%2811%29+=+%28-11%2B10%29%5E3%2B8

g%2811%29+=+%28-1%29%5E3%2B8

g%2811%29+=+-1%2B8

g%2811%29+=+7

------

g%28x%29+=+%28x%2B10%29%5E3%2B8

g%2812%29+=+%28-12%2B10%29%5E3%2B8

g%2812%29+=+%28-2%29%5E3%2B8

g%2812%29+=+-8%2B8

g%2812%29+=+0