has an inverse on a given domain if and only if there are no two distinct values on the domain  such that .

 is a quadratic function, so its graph is a parabola. The key is to find the -intercept of the vertex of the parabola, which can be found by completing the square:

The vertex happens at , so the interval which contains this value will have at least one pair  such that . The correct choice is .

 has an inverse on a given domain if and only if there are no two distinct values on the domain  such that .

 has a sinusoidal wave as its graph, with period ; it begins at a relative maximum of  and has a relative maximum or minimum every  units. Therefore, any interval containing an integer multiple of  will have at least two distinct values  such that .

The only interval among the choices that includes a multiple of  is :

 .

This is the correct choice.

If , then . By the addition property of inequality, if , then . Therefore, if , . 

Consequently, there can be no  such that , regardless of how the domain is restricted.  will have an inverse regardless of any domain restriction.

The statement "If a fish is a carnivore, then it is a shark", can be simplified to "If X, then Y", where X represents the hypothesis (i.e. "If a fish is a carnivore...") and Y represents the conclusion (i.e. "...then it is a shark").

Answer choice A is a converse statement, and not necessarily true: ("If Y, then X").

Answer choice C is an inverse statement, and not necessarily true: ("If not X, then not Y").

Answer choice D states "If not Y, then X", which is false.

Answer choice E "All fish are sharks" is also false, and cannot be deduced from the given information.

Answer choice B is a contrapositive, and is the only statement that must be true. "If not Y, then not X."

The statement given in the question suggests that all carnivorous fish are sharks. So if a fish is not a shark then it cannot be carnivorous.

With composition of functions (as with the order of operations) we perform what is inside of the parentheses first. So, g(3)=2(3)+2=8 and then f(8)=24.

First, input the function of h into g. So f(x) = 4(.25πx + 5) – 3, then simplify this expression f(x) = πx + 20 – 3 (leave in terms of π since our answers are in terms of π). Then plug in 1 for x to get π + 17.

Adding 12 to both sides and dividing by 4 yields (7y+12)/4.

A composite function substitutes one function into another function and then simplifies the resulting expression.  F(G(x)) means the G(x) gets put into F(x).

F(G(x)) = 2(x – 3)² + 3 = 2(x² – 6x +9) + 3 = 2x² – 12x + 18 + 3 = 2x² – 12x + 21

G(F(x)) = (2x² +3) – 3 = 2x²

When functions are set up within other functions like in this problem, the function closest to the given variable is performed first. The value obtained from this function is then plugged in as the variable in the outside function. Since b(x) = –2x, and x = 2, the value we obtain from b(x) is –4. We then plug this value in for x in the a(x) function. So a(x) then becomes 2(–4³) + (–4), which equals –132.