Final answer:
The function h(x) = x^3 is injective (one-to-one) because different inputs produce different outputs, surjective (onto) because every real number is an output for some input, and therefore, it is bijective (both injective and surjective).
Step-by-step explanation:
When evaluating the function h(x) = x3, we are looking to determine its injectivity, surjectivity, and bijection.
Injectivity (One-to-One)
A function is injective if different inputs produce different outputs. To determine if h(x) is injective, we assume that h(a) = h(b) implies that a3 = b3. Taking the cube root of both sides, we get that a = b, proving that h(x) is indeed injective.
Surjectivity (Onto)
A function is surjective if every element in the target set is the output of the function for at least one input from the domain. Since every real number has a real cube root, our function h(x) maps every real number to another real number, making it surjective.
Bijectivity
Since h(x) is both injective and surjective, we can conclude that the function is bijective, meaning it's a one-to-one correspondence between the elements of the domain and the range.
Therefore, the function h(x) = x3 is bijective.