Question

Mathhsssss help please

Answer 1

For the domain, you want to see what x-values (if any) are excluded from the possible values. consider: square roots always be non-negative and you cannot have 0 as the denominator.

I'll walk you through the first function as an example.

f(x) = 2x/(x^2+11x+30)

factoring the denominator gives us

\[f(x)=\frac{ 2x }{ (x+5)(x+6) }\]

your denominator must be nonzero. so x = -5 and x = -6 are both excluded from your domain. your domain is every real value except -5 and -6.

Answer 2

The range on this function is actually a bit involved to calculate by hand, so I would recommend graphing this one

If you look at the graph, you can see multiple parts where multiple domain values get assigned to the same range value. For example, for y = -4 you can see 4 possible x-values that match. So this is a many-to-one function.

Repeat this process for the other three functions.

Answer 3

a.
for range
let f(x)=y
\[y=\frac{ 2x }{ x^2+11x+30 }\]
\[x^2y+11xy+30y= 2x\]
\[yx^2+11yx-2x+30y=0\]
\[yx^2+(11y-2)x+30y=0\]
which is a quadratic in x.
it has solution \[if~discriminant \ge 0\]
\[(11y-2)^2-4y \times 30y \ge 0\]
\[121y^2-44y+4-120y^2 \ge 0\]
\[y^2-44y+4\ge 0\]
\[y^2-44y+(\frac{-44}{2})^2\ge ( \frac{-44}{2})^2-4\]
\[(y-22)^2 \ge 22^2-4\]
\[(y-22)^2\ge (484-4)\]
\[\left| y-22 \right|\ge \sqrt{480}\]
or

\[\left| y-22 \right|\ge 4\sqrt{30}\]
\[y\le 22-4 \sqrt{30}\]
or
\[y \ge 22+4\sqrt{30}\]

Answer 4

thanks for the help :)

Answer 5

Do you need help with the other functions or are you good? We can always help check your solution if you want to try it on your own.

Answer 6

lemme try myself .. next i'll need help on probability