Interval in which 'm' must lie for f(x) = x^2 + 2mx - 1 to be a one-to-one function:

To determine the interval in which 'm' must lie for the function f(x) = x^2 + 2mx - 1 to be a one-to-one function, we need to consider the properties of one-to-one functions and analyze the given function.

Properties of a One-to-One Function:
1. Every input has a unique output.
2. No two different inputs have the same output.

Analysis of the Given Function 'f(x) = x^2 + 2mx - 1':
1. The given function is a quadratic function of the form ax^2 + bx + c.
2. To determine if it is a one-to-one function, we need to analyze the discriminant of the quadratic equation.
- The discriminant, Δ, of a quadratic equation ax^2 + bx + c is given by Δ = b^2 - 4ac.
- If Δ > 0, the quadratic equation has two distinct real roots, and the function is not one-to-one.
- If Δ = 0, the quadratic equation has one real root, and the function is not one-to-one.
- If Δ < 0,="" the="" quadratic="" equation="" has="" no="" real="" roots,="" and="" the="" function="" is="" />

Analysis of the Given Function 'f(x) = x^2 + 2mx - 1':
1. The given function is a quadratic function of the form ax^2 + bx + c.
2. The quadratic coefficient, 'a', is equal to 1, as the function is written as f(x) = x^2 + 2mx - 1.
3. The linear coefficient, 'b', is equal to 2m, as the function is written as f(x) = x^2 + 2mx - 1.
4. The constant term, 'c', is equal to -1, as the function is written as f(x) = x^2 + 2mx - 1.

Determining the Discriminant:
1. The discriminant, Δ, of the given function is given by Δ = (2m)^2 - 4(1)(-1) = 4m^2 + 4.
2. For the function to be one-to-one, the discriminant Δ must be less than zero: Δ < />
- 4m^2 + 4 < />
- m^2 + 1 < />
- m^2 < -1="" (subtracting="" 4="" from="" both="" />
- Since the square of any real number is always greater than or equal to zero, there is no real number 'm' that satisfies m^2 < />

Conclusion:
There is no interval in which 'm' can lie for the function f(x) = x^2 + 2mx - 1 to be a one-to-one function. The given function will never be a one-to-one function regardless of the value of 'm'.