BELLOW!!
Explanation:
AI-generated answer
To determine if a function is one-to-one, we need to check if each input value corresponds to a unique output value. In other words, no two different input values should produce the same output value.
Let's consider the function f(x) = x^2 - 3x + k. We want to find the value of k that makes this function one-to-one.
To determine if f(x) is one-to-one, we can use the horizontal line test. This test involves drawing a horizontal line through the graph of the function. If the line intersects the graph at more than one point, then the function is not one-to-one.
The graph of f(x) = x^2 - 3x + k is a parabola. We know that a parabola is symmetric with respect to its axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola. For a parabola of the form f(x) = ax^2 + bx + c, the equation for the axis of symmetry is x = -b/2a.
In this case, the coefficient of x^2 is 1, and the coefficient of x is -3. So the axis of symmetry is x = -(-3)/(2*1) = 3/2.
If we substitute x = 3/2 into the function, we get f(3/2) = (3/2)^2 - 3(3/2) + k = 9/4 - 9/2 + k = 9/4 - 18/4 + k = -9/4 + k.
Therefore, to make the function one-to-one, we need to find the value of k such that -9/4 + k is not equal to -9/4 for any x.
Since -9/4 is a constant term, we can see that k cannot be equal to -9/4. So any value of k except -9/4 will make the function one-to-one.
Therefore, the function f(x) = x^2 - 3x + k is one-to-one for any value of k except -9/4.
In the given options, 2, 5, 11, and 13 are all different from -9/4. So any of these values for k will make the function one-to-one.