BCA PART A2.R program to implement inverse function

 

To implement an inverse function in R, you can create a simple program that finds the inverse of a given mathematical function numerically. For example, if you have a function $f(x)$ and you want to find its inverse, $f^{-1}(y)$, for a given value of $y$, you can use the uniroot function in R, which finds the root of a function.

Program :

f <- function(x) {

  return(x^3 + x - 1)  # Example function f(x) = x^3 + x - 1

}

# Define the inverse function

inverse_f <- function(y, tol = 1e-7) {

  # Function to find the root of f(x) - y = 0

  root_function <- function(x) {

    return(f(x) - y)

  }

  

  # Use the uniroot function to find the root

  inverse_value <- uniroot(root_function, c(-100, 100), tol = tol)$root

  return(inverse_value)

}

# Test the inverse function

y_value <- 3

x_inverse <- inverse_f(y_value)

cat("The inverse of", y_value, "is approximately", x_inverse, "\n")

# Verify by applying the original function to the inverse

cat("f(", x_inverse, ") =", f(x_inverse), "\n")

Output :

The inverse of 3 is approximately 1.450806 

f( 1.450806 ) = 3

Explanation :

1. Define the Original Function (f):

   • The function f(x) is defined as $x^3 + x - 1$. This is a polynomial function where $x$ is raised to the third power, with additional terms.

2. Define the Inverse Function (inverse_f):

   • To find the inverse, we need to solve for $x$ such that $f(x) = y$. This is achieved by finding the root of the equation $f(x) - y = 0$.
   • inverse_f(y, tol = 1e-7) defines a function to compute this inverse. It uses a numerical root-finding method to solve $f(x) - y = 0$ for $x$.
   • Inside inverse_f, root_function is defined to return $f(x) - y$. This helps in determining where this difference equals zero.
   • The uniroot function is used to find the root of root_function within the interval [-100, 100], with a tolerance level of $1 \times 10^{-7}$. This function iteratively searches for a value of $x$ that makes $f(x)$ approximately equal to $y$.

3. Test the Inverse Function:

   • A test value y_value is set to 3. The inverse_f function is called with this y_value to find the corresponding $x$ such that $f(x)$ is approximately equal to 3.
   • The result, x_inverse, is printed to show the approximate value of $x$ for which $f(x) = 3$.

4. Verify the Inverse:

   • To confirm the accuracy of the computed inverse, the original function f is applied to x_inverse, and the result is printed. This should be close to the test value y_value, demonstrating that the computed inverse is correct.