The two potential concerns for domain in a standard function are negative numbers inside even-powered radicals, and dividing by zero. In this case, the radical we have is odd-powered, so having a negative result underneath the radical is fine. All we need to do, then, is avoid dividing by zero, which means avoiding a result which sums to zero under the radical. In this case, only  will create this situation, so we must avoid it.

Thus, our domain is 

To find the domain, you must find the values for which you can plug in for . Based on the given function, you know that you can not have a demoninator equal to . So, by setting the denominator equal to , you can find out the values that can not be in the domain, and then simply remove them from your set of numbers from negative infinity to positive infinity.

Thus,

Therefore, for our function,  can not equal  or it will be undefined. Thus, we get the set of numbers excluding positive .

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

The key to this question is to find the zeroes of the polynomial, which can be done as follows:

'

The zeroes are . 

 has one boundary that is a zero and one interior point that is a zero. Therefore, there is a vertex in the interior of the interval, so it will have at least one pair  such that . Since a cubic polynomial has two "arms", one going up and one going down,  will increase as  increases in the other four intervals.  is the correct choice.

 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  and phase shift  units to the left. Its positive "peaks" and  "valleys" begin at  and occur every  units.

Since  includes one of these "peaks" or "valleys", it contains at least two distinct values  such that . It is the correct choice.

 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.