Podcast
Questions and Answers
What is the domain of the function $f(x) = \sqrt{x}$?
What is the domain of the function $f(x) = \sqrt{x}$?
- All non-negative real numbers (correct)
- All integers
- All real numbers
- All positive real numbers
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?
- 0
- 3 (correct)
- -3
- 1
What is the domain of the function $h(x) = \sqrt{2x - 3}$?
What is the domain of the function $h(x) = \sqrt{2x - 3}$?
- $x \leq 3/2$
- All real numbers
- $x > 0$
- $x \geq 3/2$ (correct)
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 $g(x)$?
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?
What is the domain of the function $f(x) = \sqrt{x^2 + 4}$?
What is the domain of the function $f(x) = \sqrt{x^2 + 4}$?
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?
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?
Which statement best describes the domain of a function?
Which statement best describes the domain of a function?
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)?
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?
Which of these functions has an undefined slope?
Which of these functions has an undefined slope?
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?
Flashcards
Domain of a function
Domain of a function
The set of all possible input values (x-values) for which a function is defined.
√(x-a)
√(x-a)
The domain of the square root function (√x) is all x-values such that the term inside of the square root is greater than or equal to zero (≥0).
√(2x-3)
√(2x-3)
The domain of the given function is x ≥ 3/2 (1.5).
f(x) + g(x)
f(x) + g(x)
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f(x)-g(x)
f(x)-g(x)
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f(g(x))
f(g(x))
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f(g(x)) when x = n
f(g(x)) when x = n
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f(g(1))
f(g(1))
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Domain of a Function
Domain of a Function
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Composite Function
Composite Function
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Slope of a Line
Slope of a Line
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Slope Formula
Slope Formula
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Undefined Slope
Undefined Slope
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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)
- 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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