Answer:
a): surjective (only)
b): injective (only)
c): bijective (both injective and surjective)
Explanation:
Given the functions F(x)=x(x-4)⁴, G(x)=1/x, and H(x)=x³, you want to know whether each is injective, surjective, and/or bijective.
Injective
A function is injective if its graph passes the horizontal line test. Each unique output must correspond to exactly one unique input.
Surjective
In order to determine whether a function is surjective, you must first define its "codomain," the set of all possible output values of interest. If there is an input function value corresponding to every value in the codomain, the function is surjective.
For example, the floor function produces only integer output values. It is surjective when the codomain is defined as Integers (ℤ), but not when the codomain is defined as Real numbers (ℝ).
Bijective
A function is bijective if it is both injective and surjective.
a): F(x) = x(x -4)⁴
The graph in the second attachment show this polynomial function does not pass the horizontal line test: a horizontal line may intersect the graph in more than one place. It is an odd-degree polynomial function, so can produce every possible value in ℝ, the set of real numbers. We can say ...
F(x) is not injective, is surjective on ℝ.
b): G(x) = 1/x
The function passes the horizontal line test, but cannot produce the output value 0. We can say ...
G(x) is injective, not surjective on ℝ.
c): H(x) = x³
The function is a non-decreasing odd-degree polynomial function, so it passes the horizontal line test and can produce every possible value in ℝ.
H(x) is injective, surjective, and bijective.
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Additional comment
We are familiar with the terms "domain" and "range" to describe the sets of input and output values for which a function is defined.
The "codomain" generally includes the range, along with any other output values that we may consider as possible outputs (or that we don't specifically exclude). When we say the range of G(x) is "ℝ except y=0", that function will be surjective on a codomain of {ℝ except 0}, but not on a codomain of ℝ.
Often, the only reason the codomain is different from the range is that we're lazy in describing the codomain. For example, it is easier to say "ℝ" than to say "ℝ except 0". It is easier to say "Natural numbers" than to say "even Natural numbers."