
Detailed solutions to the problems in Find the Domain of a Function  Problems are presented here.
Solution to Problem 1:

The given function is as follows
f(x) = x + 1

This a linear (polynomial) function and its domain is.
(∞ , +∞)
Solution to Problem 2:

The given function is as follows
f(x) = √(2x)

This a composed square root function. The domain is found by solving the inequality
2x ≥ 0

The solution set for the above inequality is the domain and is given by the interval
[0 , +∞)
Solution to Problem 3:

The given function is a rational function.
f(x) = (x  1) / (x  3)

Its domain is the set of all real numbers except those values of x that make the denominator zero. Hence the domain is given by the interval
(∞ , 3) U (3 , +∞)
Solution to Problem 4:

Find the domain of function f given by
f(x) = √(x + 1) / (x + 3)

To find the domain of the above function we need two conditions.

condition (1): x + 1 is under the square root and must be positive or zero. Hence
x + 1 ≥ 0 leads to x ≤ 1

condition (2): x + 3 is in the denominator and must be non zero. Hence x must not take the value 3. The two conditions must be satisfied simultaneously; hence the domain of the given function is defined by
(∞ , 3) U (3 , 1]
Solution to Problem 5:

Find the domain of function f given by.
f(x) = ^{3}√(2x + 1)

The expression 2x + 1 can take any real value. Hence the domain of the function is defined by
(∞ , +∞)
Solution to Problem 6:

The given function is
f(x) = ln (x^{2}  9)

The expression x^{2}  9 must be positive for the function to be real valued. Hence we need to solve
x^{2}  9 > 0

The above inequality can be solved by first factoring the left side.
(x  3)(x + 3) > 0

The solution set to the above polynomial inequality, which also the domain of function f, is defgined by.
(∞ , 3) U (3 , +∞)
Solution to Problem 7:

The given function is
f(x) = 2 sin(x  1)

x  1 can be any real number. Hence the domain of the above function is given by
(∞ , +∞)
Solution to Problem 8:

Find the domain of function f defined by.
f(x) = e^{(x  4)}

x  4 can take any real value and therefore the domain of f is the set of all real numbers.
Solution to Problem 9:

The given function is.
f(x) = arcsin(x^{2}  1)

For f to be real valued, the value of the expression x^{2}  1 must be restricted as follows:
1 ≤ x^{2}  1 ≤ 1 , (domain of arcsin function)

Solve the above inequality to obtain the solution set which also the domain of f.
[√(2), √(2)]
Solution to Problem 10:

Find the domain of.
f(x) = 1 / (x^{3} + x^{2} 2x)

The domain of f is restricted to those values that do not make the denominator equal to zero. Let us find the values of x that make the denominator zero.
x^{3} + x^{2} 2x = 0

Factor the left side.
x ( x^{2} + x 2) = 0

The solutions to the above equations are:
x = 0 , x = 1 and x = 2.

The domain of f is given by
(∞ , 2) U (2 , 0) U (0 , 1) U (1 , +∞)
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