Functions and domains. Please check my answers.

[need_aca_help]

Homework Statement

Question 1:
Which of the following define y as a function of x on R (Real number). Explain for each why they are/ are not function.

a) 4x^3 + y = 6
b) x - y - square root x = 8
c) x = cos^2 y
d) y = (2x + 3) / (x - 1)

Question 2:
Let g(x) = sin(x) and h(x) = 1/x be defined on their natural domains. State the following, giving the domain for each function using set notation.

a) 1 / h(x)
b) (g ∘ h)(x)
c) (h ∘ g)(x)
d) h(x)g(x)

Homework Equations

None provided.

The Attempt at a Solution

Question 1
a) y = 6 - 4x^3
Function exists

b) y = x - square root x - 8
Function exists if x = R

c) ?

d) Function exits if x = R
Function exists R \ {1}

Question 2:
a) 1 / (1/x) = x | x element R
b) g(h(x)) = sin(1/x) | x ≠ infinity
c) h(g(x)) = 1 / sin(x) | x ≠ 0
d) (1/x)(sin(x)) = sin(x) / x | x ≠ 0

OP's message:
I am having trouble understanding how to do these questions and also to write down the reasons in a mathematical way...

I have skipped the working since it will be difficult to type them. I will post a photo if necessary.
Please explain the answer if they are wrong...

 

[haruspex]

b) y = x - square root x - 8
Function exists if x = R

You mean, for any x in R? It doesn't say, but I think you're supposed to assume y must be real.

c) ?

What range of values can x have?

d) Function exits if x = R
Function exists R \ {1}

So is the answer yes or no?

Question 2:
a) 1 / (1/x) = x | x element R

1/h(x) cannot be defined at a value of x if h(x) is not defined there.

b) g(h(x)) = sin(1/x) | x ≠ infinity

You don't need to worry about x being infinity since infinity is not in R. What value of x is a problem?

c) h(g(x)) = 1 / sin(x) | x ≠ 0

What value of sin(x) is disallowed for the function 1/sin(x)? For what value(s) of x does that happen?

d) (1/x)(sin(x)) = sin(x) / x | x ≠ 0

Right.