The functions g(x) = x² and h(x) = x² represent the same mathematical expression. Here are some statements that can be evaluated for their truthfulness:
- They are both quadratic functions: True. Both functions can be classified as quadratic functions, as they can be expressed in the standard form f(x) = ax² + bx + c, where ‘a’ is not equal to zero. In this case, both g(x) and h(x) have ‘a’ equal to 1, making them quadratic.
- They have the same vertex: True. The vertex of a quadratic function in the form f(x) = ax² + bx + c is found at the point (-b/(2a), f(-b/(2a))). For both g(x) and h(x), which have no linear or constant terms, the vertex is at (0, 0).
- They will have the same graph: True. Since both functions are identical (g(x) = h(x)), their graphs will overlap perfectly, resulting in the same parabola on a coordinate plane.
- They have different ranges: False. The range of both functions is the same, as they both open upwards and the minimum value is 0 (the vertex). The range is [0, ∞).
- They are periodic functions: False. Quadratic functions are not periodic. Periodic functions repeat their values in regular intervals, whereas the values of g(x) and h(x) increase indefinitely as x moves away from the vertex in either direction.
- They intersect the x-axis at the same points: True. The functions intersect the x-axis where the output is zero, occurring at the points x = 0. Hence, both function graphs will touch the x-axis only at this point.
In summary, g(x) = x² and h(x) = x² are identical functions with several true statements regarding their characteristics, including their classification as quadratic functions, identical graphs, and shared properties such as the vertex and intersection points on the x-axis.