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Questions and Answers
What is the domain of the function $f(x) = \sqrt{x - 3}$?
What is the domain of the function $f(x) = \sqrt{x - 3}$?
- $[0, 3]$
- All real numbers
- $(-\infty, 3)$
- $[3, \infty)$ (correct)
For the function $g(x) = \sqrt{x + 1}$, what is the minimum value of $x$ in its domain?
For the function $g(x) = \sqrt{x + 1}$, what is the minimum value of $x$ in its domain?
- 2
- -1 (correct)
- 0
- 1
What is the domain of the function $h(x) = \frac{1}{x - 2}$?
What is the domain of the function $h(x) = \frac{1}{x - 2}$?
- $(-\infty, 2)$
- All real numbers except 2 (correct)
- $[0, 2)$
- $(-\infty, 0) \cup (0, 2)$
For the function $f(x) = \frac{1}{\sqrt{x}}$, which of the following statements is true about its domain?
For the function $f(x) = \frac{1}{\sqrt{x}}$, which of the following statements is true about its domain?
What is the domain for the function $j(x) = \sqrt{2x - 3}$?
What is the domain for the function $j(x) = \sqrt{2x - 3}$?
Given the function $f(x) = x^2 - 4$, what values of $x$ will yield a positive output?
Given the function $f(x) = x^2 - 4$, what values of $x$ will yield a positive output?
What is the range of the function $k(x) = \sqrt{x + 2}$?
What is the range of the function $k(x) = \sqrt{x + 2}$?
For the function $m(x) = x^2 + 3x + 2$, which interval represents the real solutions of $m(x) = 0$?
For the function $m(x) = x^2 + 3x + 2$, which interval represents the real solutions of $m(x) = 0$?
Flashcards
Domain of a Function
Domain of a Function
The set of all possible input values (x-values) for which a function is defined.
Square Root Function Domain
Square Root Function Domain
The input values (x-values) for which the expression inside the square root symbol (radical) is non-negative.
Finding Function Composition
Finding Function Composition
Substituting the function expression (g(x)) into the function (f(x)) and simplify. This is the process of combining two functions into a single function where the output of the first function becomes the input to second. Write as f(g(x)).
f(g(x))
f(g(x))
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f(g(a))
f(g(a))
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f(-2)
f(-2)
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f(g(1))
f(g(1))
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f + g(x)
f + g(x)
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f - g(x)
f - g(x)
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Study Notes
Function Domains
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Finding Domains: Determine the set of all possible input values (x-values) for a function where the output is defined.
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Restrictions: Look for values that would result in division by zero, or the square root of a negative number; these are not permitted.
Example Problems
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Problem 1: f(x) = (x + 3)² / (x - 3).
- Domain: All real numbers except x = 3.
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Problem 2: f(x) = √(8 - x).
- Restriction: 8 - x ≥ 0.
- Domain: All real numbers less than or equal to 8, or (-∞, 8].
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Problem 3: f(x) = √(2x - 3).
- Restriction: 2x - 3 ≥ 0.
- Domain: x ≥ 3/2, or [3/2, ∞).
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Problem 4: f(x) = (x² - 2x - 3).
- Restriction: x² - 2x - 3 ≠0.
- Factor x² - 2x - 3 as (x - 3)(x + 1).
- Solution: x ≠3 and x ≠-1.
- Domain: All real numbers except 3 and -1. (-∞, -1) U (-1, 3) U (3, ∞).
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Problem 5: f(x) = (x + 4)/(√x + 2).
- Restriction: x + 4 exists, x + 2 is greater than 0.
- Domain x ≥ -2.
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Problem 6: Functions involving f(x) = x - 3, g(x) = √(x + 1).
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f + g(x) = x - 3 + √(x + 1)
- Domain: x ≥ -1
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f − g(x) = x - 3 - √(x + 1)
- Domain: x ≥ -1
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f ⋅ g(x) = (x - 3)√(x + 1)
- Domain: x ≥ -1
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g/f
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Domain: x ≥ -1, x ≠3.
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Combining Functions
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Sum, difference, product, and quotient of functions: Find the domain by combining the restrictions of the individual functions .
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Composite Functions (f ∘ g)(x) = f(g(x)): Determine the domain of f(g(x)) by analyzing the domain of the inner function (g(x)) and the restriction of g on the domain of f.
Example Composite Function
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Problem 1. Given f(x) = x + 3 and g(x) = √(x + 2).
- (f ∘ g)(x) = f(g(x)) = f(√(x + 2)) = √(x + 2) + 3
- Restriction: x + 2 ≥ 0, or x ≥ -2.
- (f ∘ g)(x) = f(g(x)) = f(√(x + 2)) = √(x + 2) + 3
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Problem 2: Given f(x) = x + 3 and g(x) = √(x + 2); Find the domain of g(f(x)).
- g(f(x)) = √(x + 3 + 2) = √(x + 5)
- Restriction: x + 5 ≥ 0, or x ≥ -5.
- Domain: [-5, ∞).
- g(f(x)) = √(x + 3 + 2) = √(x + 5)
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