Alan

P(n) = 4ⁿ + 15n - 1.       P(1) = 18.   Divisible by 9

P(n+1) = P(n) + Q(n).      (A)         where Q(n) = 3*4ⁿ + 15

Q(n) = 3*4ⁿ + 15.    Q(1) = 27. Divisible by 9

Q(n+1) = 3*4ⁿ⁺¹ + 15.    Q(n+1) = 12*4ⁿ + 15  Q(n+1) = 9*4ⁿ + Q(n).

Hence if Q(n) divisible by 9 then Q(n+1) divisible by 9 

Hence Q(n) divisible by 9 (because Q(1) also divisible by 9)

So if P(n) divisible by 9 then P(n+1) divisible by 9 (because both terms on RHS of A are divisible by 9).

Since P(1) is also divisible by 9 then P(n) is divisible by 9.  QED.

This graph might give you a clue:

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Simply replace the x's in f(x) by 5/(x - 1) 

\(f(g(x))=\frac{5/(x-1)}{5/(x-1)+3}\rightarrow \frac{5}{5+3(x-1)}\rightarrow \frac{5}{3x+2}\)

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"A function has zeroes at -1, 3, and 5. We know that f(-2) and f(2) are negative, while f(4) and f(6) are positive. Sketch a graph of f(x)."

One possibility:

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Here are some of the intermediate steps Melody:

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Look at the values near t = 0.

At t = 0, cos(0) is +1, and since the graph shows y₂(0) is positive, then a₂ must be positive.

For small values of b₁t, sin(b₁t) ≈ b₁t.  For b₁ positive, this is positive, and since the graph shows y₁(t) positive for small values of t, then a₁ must be positive.

a₁ and a₂ represent the amplitudes of y₁ and y₂ respectively.  Since it is clear from the graph that the amplitude of y₁ is greater than that of y₂, we must have a₁>a₂.

Summarising:    0 < a₂ < a₁

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_.

Just looked through a few on-line American dictionaries.  They all use radii for the plural of radius.

"What is the floor value of       -6.5                   [Or any number between -7 and -6]

I assume it is -7, is that correct ?"

Correct!

"Given any positive even integer, x, the positive difference between the smallest odd number greater than 7x-2  and the largest odd number less than 3x+5 can be written in the form ax+b . What is a+b?"

If x is even then 7x - 2 is even, so the smallest odd number greater than 7x - 2 is 7x - 2 + 1 or 7x - 1

With x even then 3x + 5 is odd, so the largets odd number less than 3x + 5 is 3x + 5 - 2 or 3x + 3

Difference:  7x - 1 - (3x + 3) = 4x - 4  

ax + b = 4x - 4  so a = 4, b = -4

a + b = 0.

"Suppose f and g are polynomials, and that h(x)=f(g(x))+g(x). Find the degree of g(x) given that the degree of h(x) is 8 and the degree of f(x) is 4."

f(x) = a*x⁴ + b*x³ + ...

g(x) = c*xⁿ + d*x^n-1 + ...

h(x) = a*(c*xⁿ + d*x^n-1 + ...)⁴ + b*(c*xⁿ + d*x^n-1 + ...)³ + ... + c*xⁿ + d*x^n-1 + ...

h(x) = a*c⁴*x⁴ⁿ + ....

Must have 4n = 8  hence n = 2

Degree of g(x) is 2.