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Questions and Answers
What is the domain of the function $f(x) = \sqrt{x}$?
What is the domain of the function $f(x) = \sqrt{x}$?
For the function $g(x) = \frac{1}{x-3}$, which value makes the function undefined?
For the function $g(x) = \frac{1}{x-3}$, which value makes the function undefined?
What is the domain of the function $h(x) = \sqrt{2x - 3}$?
What is the domain of the function $h(x) = \sqrt{2x - 3}$?
If $f(x) = x + 3$ and $g(x) = \sqrt{x + 2}$, what is the domain of $g(x)$?
If $f(x) = x + 3$ and $g(x) = \sqrt{x + 2}$, what is the domain of $g(x)$?
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For the function $f(x) = \frac{x}{x^2 - 4}$, which values must be excluded from the domain?
For the function $f(x) = \frac{x}{x^2 - 4}$, which values must be excluded from the domain?
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What is the domain of the function $f(x) = \sqrt{x^2 + 4}$?
What is the domain of the function $f(x) = \sqrt{x^2 + 4}$?
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If $f(x) = 5$ and $g(x) = x^2 - 9$, for which values does $g(x)$ yield a negative result?
If $f(x) = 5$ and $g(x) = x^2 - 9$, for which values does $g(x)$ yield a negative result?
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Which of the following functions has a restricted domain due to a square root?
Which of the following functions has a restricted domain due to a square root?
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Which statement best describes the domain of a function?
Which statement best describes the domain of a function?
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What happens to the domain of a composite function (f ∘ g)(x) if g(x) outputs a value outside the domain of f(x)?
What happens to the domain of a composite function (f ∘ g)(x) if g(x) outputs a value outside the domain of f(x)?
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In the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, what does 'm' represent?
In the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, what does 'm' represent?
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Which of these functions has an undefined slope?
Which of these functions has an undefined slope?
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If f(x) = x + 1 and g(x) = 1/(x - 2), which of the following statements about the composite function (f ∘ g)(x) is true?
If f(x) = x + 1 and g(x) = 1/(x - 2), which of the following statements about the composite function (f ∘ g)(x) is true?
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Study Notes
Domain of Functions
- Finding the domain: Determine the set of all possible input values (x-values) for a function.
Problem 1
- Function: f(x) = (x + 3)² / (x² − 3)
- Domain: All real numbers except x = ±√3 (where the denominator is zero)
Problem 2
- Function: f(x) = √(8 - x) / (x - 10)
- Domain: x ≤ 8 and x ≠ 10
Problem 3
- Function: f(x) = √(x + 3) / (x + 4)
- Domain: x ≥ -3 and x ≠ -4
Problem 4
- Function: f(x) = √(2x - 3)
- Domain: x ≥ 3/2
Problem 5
- Function: f(x) = x² - 2x - 3
- Domain: All real numbers (since it's a quadratic without restrictions)
Problem 6 and 7
- Composition of functions f(x) = x - 3, g(x) = √(x + 1)
- f + g(x): (x - 3) + √(x + 1)
- f - g(x): (x - 3) - √(x + 1)
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Domain restrictions: x ≥ -1 for g(x)
- f + g(x) Domain: x ≥ -1
- f - g(x) Domain: x ≥ -1
Problem 8
- Composition of functions.
- f + g(x): x - 3 + √(x + 1)
- f - g(x): x - 3 - √(x + 1)
- Domain restrictions: x ≥ -1 for all compositions
Problem 9
- f(x) = x - 3, g(x) = √(x + 1)
- f(x) + g(x): x − 3 + √(x + 1)
- Domain: x ≥ -1 for √(x + 1)
Problem 10
- f(16): Given f(x) = x - 3, g(x) = √(x + 1)
- f + g(16): 16 − 3 + √(16 + 1) = 13 + √17
- Domain: x ≥ -1, so f(x) and g(x) are defined
Problem 11 and 12 and 13
- Function compositions: Calculate g(f(x)) by substituting f(x) into g(x), g(f(12)), and finding f – g(-2) and fg(1). Details are included in given data.
Problem 14
- Function composition: f(x) = x + 3, g(x) = √(x + 2)
- f ◦ g(x): f(√(x + 2)) = √(x + 2) + 3
- Domain: x ≥ -2
Problem 15
- Function composition: f(x) = x + 3, g(x) = √(x + 2)
- g ◦ f(x): g(x + 3) = √(x + 3 + 2) = √(x + 5)
- Domain: x ≥ -5
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Description
Test your understanding of the domain of functions with this quiz. You will be required to determine the domain for various functions, including those with square roots and rational expressions. This quiz covers composition as well as basic domain restrictions.
