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Detailed solutions to the problems in Find the Domain of a Function - Problems are presented here.
Solution to Problem 1:
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The given function is as follows
f(x) = x + 1
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This a linear (polynomial) function and its domain is.
(-∞ , +∞)
Solution to Problem 2:
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The given function is as follows
f(x) = √(2x)
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This a composed square root function. The domain is found by solving the inequality
2x ≥ 0
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The solution set for the above inequality is the domain and is given by the interval
[0 , +∞)
Solution to Problem 3:
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The given function is a rational function.
f(x) = (x - 1) / (x - 3)
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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:
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Find the domain of function f given by
f(x) = √(-x + 1) / (x + 3)
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To find the domain of the above function we need two conditions.
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condition (1): -x + 1 is under the square root and must be positive or zero. Hence
-x + 1 ≥ 0 leads to x ≤ 1
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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:
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Find the domain of function f given by.
f(x) = 3√(2x + 1)
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The expression 2x + 1 can take any real value. Hence the domain of the function is defined by
(-∞ , +∞)
Solution to Problem 6:
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The given function is
f(x) = ln (x2 - 9)
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The expression x2 - 9 must be positive for the function to be real valued. Hence we need to solve
x2 - 9 > 0
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The above inequality can be solved by first factoring the left side.
(x - 3)(x + 3) > 0
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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:
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The given function is
f(x) = 2 sin(x - 1)
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x - 1 can be any real number. Hence the domain of the above function is given by
(-∞ , +∞)
Solution to Problem 8:
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Find the domain of function f defined by.
f(x) = e(x - 4)
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x - 4 can take any real value and therefore the domain of f is the set of all real numbers.
Solution to Problem 9:
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The given function is.
f(x) = arcsin(x2 - 1)
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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)
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Solve the above inequality to obtain the solution set which also the domain of f.
[-√(2), √(2)]
Solution to Problem 10:
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Find the domain of.
f(x) = 1 / (x3 + x2 -2x)
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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
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Factor the left side.
x ( x2 + x -2) = 0
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The solutions to the above equations are:
x = 0 , x = 1 and x = -2.
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The domain of f is given by
(-∞ , -2) U (-2 , 0) U (0 , 1) U (1 , +∞)
More Math Problems, Questions and Online Self Tests.
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