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 (x² - 9)
  
• The expression x² - 9 must be positive for the function to be real valued. Hence we need to solve
  x² - 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(x² - 1)
  
• For f to be real valued, the value of the expression x² - 1 must be restricted as follows:
  -1 ≤ x² - 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 / (x³ + x² -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.
  x³ + x² -2x = 0
  
• Factor the left side.
  x ( x² + 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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