What input value produces the same output value for the two functions represented in the equation x³ – 2x – 3?

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Finding Input Values for Function Matches

The question involves identifying an input value that returns the same output for two distinct functions. In this case, we are considering the function f(x) = x³ – 2x – 3.

To solve the problem, we can analyze the equation:

x³ - 2x - 3 = 0

This cubic equation can be solved using various methods, such as the Rational Root Theorem, synthetic division, or numerical methods like Newton’s method if necessary.

Steps to Solve the Cubic Equation

  1. Identify Possible Roots: Start by checking if there are rational roots by substituting values of x (±1, ±2, etc.) into the function.
  2. Use Synthetic Division: If a rational root is identified, use synthetic division to factor the cubic polynomial into a quadratic equation.
  3. Quadratic Formula: If necessary, apply the quadratic formula to find the remaining roots of the quadratic equation.
  4. Confirm Roots: Once potential solutions are found, substitute these back into the original function to verify that they yield the same output.

Example

To illustrate, let’s evaluate the function for various integer values:

f(1) = 1³ - 2(1) - 3 = -4
f(2) = 2³ - 2(2) - 3 = 1
f(-1) = (-1)³ - 2(-1) - 3 = 0

When x = -1, the output is 0. This indicates that f(-1) is a solution to f(x) = 0. From this process, you can find any intersections or equivalences between your function and others you may be dealing with.

Conclusion

The key takeaway is that to find the input value that gives the same output for multiple functions, analyzing the functional equations and utilizing algebraic methods is crucial. Whether through scrutiny of rational roots or through precise numerical methods, determining the common values will allow successful resolution of the initial inquiry.

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