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)

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) (Show Source): You can put this solution on YOUR website!

If f(x) is one-to-one, then implies that 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.

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If , then...

Substitution

Subtract 10 from both sides

Multiply both sides by -1

Cube both sides

Add 8 to each side

So we end up with after assuming

So this means that leads to . If we follow the steps shown above in reverse, then we'll go from to

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

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

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

Replace f(x) with y

Swap x and y. From here on out, we're solving for y.

Subtract 10 from both sides

Multiply both sides by -1

Cube both sides

Add 8 to both sides

The inverse is

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

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SOLUTION: Analytically show that the function f(x)=10-&#8731;(x-8) is one-to-one, find its inverse, and evaluate the following: f^(-1) (10) f^(-1) (11) f^(-1) (12)

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)