Final answer:
To make the quadratic function f(x) = 3x² - 4 invertible, the domain must be restricted to either side of the vertex. Options a) x ≥ 0 or b) x ≤ 0 will create a one-to-one function.
Step-by-step explanation:
For the function f(x) = 3x² - 4, to be invertible, we need the function to be one-to-one on its domain, meaning each value of f(x) is paired with exactly one value of x. A quadratic function is naturally not one-to-one because it is symmetric about its vertex. Since f(x) opens upwards, the function will be one-to-one if we restrict the domain to either the side to the right of the vertex (x ≥ 0) or the side to the left of the vertex (x ≤ 0).Choosing either a) x ≥ 0 or b) x ≤ 0 will satisfy this requirement. Restrictions c) x ≠ 0 and d) x > 0 do not include the vertex, so they are not sufficient to ensure the function is one-to-one. Therefore, the correct answers are either a) x ≥ 0 or b) x ≤ 0.
To determine which domain restrictions will allow f(x) to be invertible, we need to consider the graph of the function. The function f(x) = 3x² - 4 is a quadratic function, and it is invertible if and only if it passes the horizontal line test. The horizontal line test states that a function is invertible if no horizontal line intersects the graph of the function more than once.In the case of f(x) = 3x² - 4, the graph is a parabola that opens upwards. This means that the function is not invertible for all real numbers, as there are horizontal lines that intersect the parabola more than once. Therefore, the correct domain restriction to make f(x) invertible would be d) x > 0. This restriction ensures that the parabola is restricted to the positive x-axis and no horizontal line would intersect it more than once.