• need_aca_help
In summary: This can be written as g(x)h(x).In summary, the first question asks to define y as a function of x on the set of real numbers for four different equations and determine if they are functions or not. The second question involves using set notation to define the domains of four compositions of two given functions. Clarification and explanations are requested for some of the 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)

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.

need_aca_help said:
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.

1 person

## What is a function?

A function is a mathematical relationship between two quantities, where each input (domain) has only one output (range).

## What is a domain?

The domain of a function is the set of all possible input values for which the function is defined.

## What is the range of a function?

The range of a function is the set of all possible output values that result from the input values in the domain.

## How do you determine the domain of a function?

To determine the domain of a function, you need to look at the restrictions on the input values. The domain can be determined by looking at the type of function (e.g. linear, quadratic, trigonometric) and any given constraints (e.g. division by zero, square root of a negative number).

## Can a function have multiple domains?

No, a function can only have one domain. However, it is possible to have a piecewise function, which has different rules for different intervals of the domain.

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