Knowing one term of a sequence will not help you find any other terms, so neither statement alone will answer the question. But knowing two terms and knowing that the sequence is arithmetic will allow you to find the common difference $\displaystyle d$. 

The twentieth term is $\displaystyle 16d$ greater than the fourth term, so take the difference and divide by 16:

$\displaystyle d = \frac{418-50}{16} = 23$

Now solve this inequality for $\displaystyle t$ to find the minimum number of terms needed to exceed 1,000:

$\displaystyle 50 + 23t \geq 1,000$

$\displaystyle a \bullet b = (a^{2} +1) (b- 10)$ , and $\displaystyle a ^{2} + 1 \geq 0 + 1 = 1 > 0$. Therefore, since $\displaystyle a ^{2} + 1$ is a positive number, the sign of $\displaystyle b - 10$ is the sign of $\displaystyle a \bullet b$. This makes Statement 1 neither necessary nor helpful. We need to know whether $\displaystyle b - 10$ is greater than, equal to, or less than 0, or, equivalently, whether $\displaystyle b$ is greater than, equal to, or less than 10. Statement 2 does not tell us this either. 

Statement (1) does not give us any information about g(x), so it is not sufficient.

Statement (2) alone gives us the relationship between the functions f and g but does not give us any information about f(x), so it is not sufficient.

Both statement together allow us to get an expression for both f and g:

$\displaystyle f(x) = 3x - 1$

$\displaystyle g(x)=(f(x))^{2}=(3x-1)^{2}=9x^{2}-6x+1$

With an expression for both functions we can estimate f(g(1)):

$\displaystyle g(1)=9-6+1=4$ and $\displaystyle f(4)=3\cdot4-1=11$

So the correct answer is C.

A necessary and sufficient condition for $\displaystyle f$ not to have an inverse is for $\displaystyle f(a) = f(b)$ for distinct $\displaystyle a,b$ in the domain of $\displaystyle f$.

If $\displaystyle f(4)- f(-4)= 0$, then $\displaystyle f(4)=f(-4)$, meaning that $\displaystyle f$ pairs at least two domain values, 4 and $\displaystyle -4$, with the same range value. Therefore, from Statement 1 alone, it follows that $\displaystyle f$ does not have an inverse.

If $\displaystyle f(7) +f(-7)= 0$, then $\displaystyle f(7) = -f(-7)$, meaning $\displaystyle f$ pairs two different domain values with two different range values. This is nt a contradiction of the conditions for an inverse to not exist, but this does not prove that an inverse does exist. Statement 2 is unhelpful either way.

Statement 1 is simply an assertion that the relation passes the vertical line test and is therefore a function. Since we know already that the relation is a function, this statement is unhelpful.

If Statement 2 is assumed, it follows that the graph passes the horizontal line test for the existence of an inverse. Statement 2 alone answers the question.

The relation comprises ten ordered pairs. If Statement 1 alone is known, then the domain comprises ten elements, each of which must appear in exactly one ordered pair. Therefore, no domain element is matched with more than one range element, and the relation is a function. 

If Statement 2 alone is known, then the range comprises ten elements, each of which must appear in exactly one ordered pair. But nothing is known about the domain. If no domain element is repeated among the ordered pairs, the relation is a function; otherwise, the relation is not a function.

$\displaystyle (fg)(1) = f(1) \cdot g(1)$. So, for $\displaystyle (fg)(1)$ to be negative, $\displaystyle f(1)$ and $\displaystyle g(1)$ must be of unlike sign.

From Statement 1 it can be determined that $\displaystyle f(1)$ is negative, but no information about the sign of $\displaystyle g(1)$ can be determined.

From Statement 2, it can be determined that $\displaystyle g(1) = -1^{2} = -1$ and is therefore negative,  but no information about the sign of $\displaystyle f(1)$ can be determined.

From the two statements together, both $\displaystyle f(1)$ and $\displaystyle g(1)$ can be proved negative, so their product, $\displaystyle f(1) \cdot g(1) = (fg)(1)$, is positive. 

$\displaystyle \left ( f \circ g\right )(M) = f (g (M))$, so we need to determine whether $\displaystyle f (g (M)) > 0$.

From Statement 1 alone, since the range of $\displaystyle f$ is $\displaystyle (0, \infty )$ - that is, the set of all positive numbers, then regardless of the value of $\displaystyle g(M)$, 

$\displaystyle \left ( f \circ g\right )(M) = f (g (M)) > 0$.

Therefore, Statement 1 alone yields an affirmative answer to the question.

From Statement 2 alone, regardless of the value or $\displaystyle M$, $\displaystyle g(M) < 0$, but we do not know the value or range of values of $\displaystyle f(g(M))$. Statement 2 alone is unhelpful.

The two statements together do not prove or disprove the relation to be a function. 

The relations defined by the sets of points

$\displaystyle \left \{ (1,1), (2,2), (3,3), (4,4), (5,5) \right \}$.

and

$\displaystyle \left \{ (1,1), (1,2), (2,2), (3,3), (4,4), (5,5) \right \}$

have the domain and range given in the statements, but the former is a function, since each domain element is matched with exaclty one element, and the latter is not a function, since domain element 1 is matched with two different range elements.

If Statement 1 alone is assumed, then, since there are only six domain elements and ten points in the relation, at least one of the domain elements must match with more than one range element. This forces the relation to not be a function.

If Statement 2 alone is assumed, then, since $\displaystyle x= 5$ is a vertical line that passes through the graph twice, the relation fails the vertical line test and is therefore not a function.