Troubleshooting Inverse Functions: Solving Algebraic Problems

In summary, The conversation is about a person trying to solve an inverse function problem involving the equation y = cuberoot(x+sqrt(1+x^2)) + cuberoot(x-sqrt(1+x^2)). They first try replacing x with y and solving for y, but get stuck when simplifying the equation. They then receive a tip to use (a+b)³ = a³ + b³ +3ab(a+b), which leads them to the correct answer of y=(1/2)(3x+x^3). They reflect on how they used to be able to solve these types of problems easily, but now they feel like a 9th grader again.
  • #1
aquitaine
30
9
Ok, I decided to review basic algebra since I haven't done anything with it in like, forever. I came across an inverse function problem that I can't get the right answer.

the equation is:

y = cuberoot(x+sqrt(1+x^2)) + cuberoot(x-sqrt(1+x^2))

I tried replacing X with Y, and solving for Y
and getting rid of the cube roots by cubing both sides
X^3 = y + sqrt(1+y^2) + y - sqrt(1+y^2)
simplifying a bit (the square roots go away)
x^3 = 2y
so
y = (1/2)x^3

Yet the book I'm using says the answer is y=(1/2)(3x+x^3)

What did I do wrong?
 
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  • #2
(a+b)³ =/= a³ + b³, as you have done.

Use (a+b)³ = a³ + b³ +3ab(a+b).

Start with y = a+b, and cube both sides. If you put a+b=y again, the result comes quickly,
 
  • #3
thanks, I think I got it. Man, a few years ago I could have done this, but now? I feel like a 9th grader again.
 

FAQ: Troubleshooting Inverse Functions: Solving Algebraic Problems

What is the definition of an inverse function?

An inverse function is a function that undoes the action of another function. It swaps the input and output values of the original function.

How do you know if two functions are inverses of each other?

To determine if two functions are inverses, you can use the horizontal line test. If a horizontal line intersects the graph of both functions at most once, then the functions are inverses of each other.

What is the process for finding the inverse of a function?

To find the inverse of a function, you can follow these steps:

  1. Write the function in the form of y = f(x).
  2. Swap the x and y variables, so the equation becomes x = f(y).
  3. Solve for y to get the inverse function, y = f-1(x).

How do you graph an inverse function?

To graph an inverse function, you can use the original function's graph and reflect it across the line y = x. This means that any points (x,y) on the original graph will become (y,x) on the inverse graph. You can also use a table of values to plot points for the inverse function.

What are some common mistakes when solving algebraic problems involving inverse functions?

Some common mistakes when solving algebraic problems involving inverse functions include:

  • Forgetting to swap the x and y variables when finding the inverse function.
  • Not using parentheses when substituting values into the inverse function, leading to incorrect calculations.
  • Using the wrong inverse function for the given original function.
  • Not checking the domain and range restrictions for both the original and inverse functions.

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