Final answer:
The function G(x)=5x+3, for all x in Z, is both injective (since different inputs produce different outputs) and surjective (since every element in Z can be mapped from an element in Z). Therefore, G(x) is bijective.
Step-by-step explanation:
The function G(x) = 5x+3 defined from the set of integers Z to itself (Z) can be evaluated for the properties of injectivity, surjectivity, and bijectivity as follows:
Injectivity (One-to-One)
A function is injective (one-to-one) if different inputs produce different outputs. In other words, if a and b are elements of the domain Z, and G(a) = G(b), then a must equal b. For G(x), suppose G(a) = G(b), then 5a + 3 = 5b + 3. Subtracting 3 from both sides yields 5a = 5b, and dividing by 5 gives a = b. Hence, G(x) is injective.
Surjectivity (Onto)
A function is surjective (onto) if every element y in the codomain Z has at least one x in the domain such that G(x) = y. Given any y in Z, we can solve the equation 5x + 3 = y for x, yielding x = (y - 3)/5, which is also an integer because the difference of two integers is an integer and an integer divided by 5 is an integer if the integer was a multiple of 5 to begin with. Thus, any integer y can be expressed as 5 times some other integer plus 3, showing that G(x) is surjective.
Bijectivity
Since G(x) is both injective and surjective, it is by definition bijective. A bijective function is one that is one-to-one and onto, which means it has an inverse. Therefore, G(x) is bijective.