What is the inverse function of x^3 + 1 and how do you find its value at x = 4?

In summary, the conversation discussed the concept of inverse functions and how to find the inverse of a given function. The problem at hand was to find the value of x that satisfies f(x) = 4, given f(x) = x^3 + 1. While one person got an answer of 0.02 by finding the reciprocal of 64, the correct approach was to find the inverse function and plug in the given value. The correct answer was found to be approximately 1.44.
  • #1
UrbanXrisis
1,196
1
I was just doing a problem in a review book and my answer doesn't match with what the books says.

It f(x)=x^3+1 and if f^-1 is the inverse function of f, what is f^-1(4)?

I got an answer of 0.02 but the book says it is 1.44.

All I did was sub in 4 into the equation, which I get 64, then set that under one...1/64 and get 0.02 as the inverse. How did the book get 1.44?
 
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  • #2
You're mistaking the inverse with the reciprocal.

The problem is asking for what value of x will satisfy f(x) = 4.

Clearly f(.002) does not equal 4, but 4(1.44) = 1.44^3 + 1 = 3.985984 ~ 4

cookiemonster
 
  • #3
The notation is what really confuses people most of the time. The inverse of a function basically switches the x and y values. If you want to find the inverse for a function than all you would have to do is switch the values. In other words, replace all the x's with y's and vice versa. Than solve for y (assuming you originally had a y= equation) and you will have your inverse function. Than you can just plug your value in. In this case it would be:
y = x^3 + 1
For the inverse:
x = y^3 + 1
y = (x - 1) ^ (1/3)
so f^-1(4) = (4 - 1) ^ (1/3) = 3 ^ (1/3) ~ 1.44
 

1. What is the inverse function of x^3 + 1?

The inverse function of x^3 + 1 is (x - 1)^(1/3).

2. How do you find the inverse function of x^3 + 1?

To find the inverse function of x^3 + 1, you need to solve for x in terms of y. This can be done by following the steps of finding the inverse of any function, which includes switching the x and y variables and solving for y.

3. What is the domain and range of the inverse function of x^3 + 1?

The domain of the inverse function (x - 1)^(1/3) is all real numbers, since any value of x can be cubed and then have 1 added to it. The range is also all real numbers, as the cube root of any real number is another real number.

4. Is the inverse function of x^3 + 1 a one-to-one function?

Yes, the inverse function of x^3 + 1 is a one-to-one function because each input has only one corresponding output and vice versa. This can also be seen through the horizontal line test, where no horizontal line can intersect the graph of the inverse function more than once.

5. What is the graph of the inverse function of x^3 + 1?

The graph of the inverse function of x^3 + 1 is a reflection of the graph of x^3 + 1 about the line y = x. This means that the inverse function will have a point at (0,1) and will approach infinity as x approaches negative infinity or positive infinity.

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