Domain of Functions

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What is the domain of a function?
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The domain of a function is the set of all possible input values (usually x-values) that the function can accept.
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How do you determine the domain of a simple polynomial function?
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For a simple polynomial function, the domain is all real numbers because polynomials are defined for all x-values.
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What is the domain of the function f(x) = 1/x?
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The domain of f(x) = 1/x is all real numbers except x ≠ 0, since division by zero is undefined.
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What changes in the domain for a function f(x) = √x?
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The domain for f(x) = √x is all x ≥ 0, since the square root of a negative number is not defined in real numbers.
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How do you find the domain of a rational function like f(x) = (x+1)/(x-2)?
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To find the domain of a rational function, set the denominator not equal to zero. For f(x) = (x+1)/(x-2), the domain is all x ≠ 2.
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Explain how logarithms affect the domain of a function.
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The domain of a logarithmic function like f(x) = log(x) is x > 0, as logarithms are undefined for zero and negative numbers.
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If a function f(x) is defined by f(x) = 1/(x^2 - 4), what is its domain?
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For f(x) = 1/(x^2 - 4), find the values that make the denominator zero: x^2 - 4 = 0, x = ±2. Thus, the domain is all x ≠ ±2.
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What is the implication of having a square root in the denominator of a function?
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If there is a square root in the denominator, the expression inside the square root must be greater than zero.
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How do you determine the domain of a piecewise function?
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Identify the domain for each piece separately, based on their individual expressions and conditions, then combine them.
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For the function f(x) = √(x-3), what restriction does the square root impose on the domain?
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The domain of f(x) = √(x-3) is x ≥ 3, because the expression inside the square root must be non-negative.
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How does an absolute value affect the domain of a function?
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An absolute value function typically does not impose domain restrictions unless paired with other operations that do.
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If a function has a fraction with a variable in the denominator, what must you exclude from the domain?
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You must exclude values that make the denominator zero, as division by zero is undefined.
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For a composite function like h(x) = sqrt(x - 5)/(x + 3), how do you find the domain?
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For h(x) = sqrt(x - 5)/(x + 3), determine where x - 5 ≥ 0 (x ≥ 5) and x + 3 ≠ 0 (x ≠ -3). The domain is x ≥ 5, x ≠ -3.
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What strategy can be used to determine the domain of a function with multiple operations?
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Identify domain restrictions for each operation separately (e.g., no zero denominator, no negatives under even roots) and combine them.
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Can the domain of a function ever be written in interval notation?
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Yes, the domain can be expressed in interval notation indicating the range of x-values that are acceptable for the function.
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