What is the inverse of the function f(x) = 2x + 1/x?

by
f(x) = 2x + 1/x, we follow a systematic approach. The steps to find the inverse function can be outlined as follows:

  1. Replace f(x) with y: This gives us the equation y = 2x + 1/x.
  2. Swap x and y: We switch the roles of x and y. The equation then becomes x = 2y + 1/y.
  3. Multiply both sides by y: To eliminate the fraction, we multiply through by y, resulting in xy = 2y^2 + 1.
  4. Rearrange the equation: We can rearrange the equation to form a standard quadratic equation: 2y^2 – xy + 1 = 0.
  5. Use the quadratic formula: To solve for y, we can apply the quadratic formula, which states that if we have an equation of the form ay^2 + by + c = 0, the solutions can be found using y = (-b ± √(b² – 4ac)) / (2a). Here, a = 2, b = -x, and c = 1.
  6. Substituting into the quadratic formula: We substitute the values into the quadratic formula: y = (x ± √(x² – 8)) / 4.

Thus, the inverse function of f(x) = 2x + 1/x can be expressed in two parts, depending on the sign chosen in the quadratic formula:

  • f-1(x) = (x + √(x² – 8)) / 4
  • f-1(x) = (x – √(x² – 8)) / 4

It’s important to consider the domain and range of the original function to choose the correct part of the inverse function, ensuring it matches the expected values in its application.

Leave a Comment