Solutions to Domain of a Function Problems

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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