# 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 , +∞)