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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 $f(x) = \sqrt{x - 3}$?
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?
What is the domain of the function $h(x) = \frac{1}{x - 2}$?
What is the domain of the function $h(x) = \frac{1}{x - 2}$?
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?
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What is the domain for the function $j(x) = \sqrt{2x - 3}$?
What is the domain for the function $j(x) = \sqrt{2x - 3}$?
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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?
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What is the range of the function $k(x) = \sqrt{x + 2}$?
What is the range of the function $k(x) = \sqrt{x + 2}$?
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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 $m(x) = 0$?
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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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Description
Test your understanding of how to determine the domains of various functions. This quiz covers concepts like identifying restrictions due to division by zero and square roots of negative numbers. Solve the problems to find the set of all possible input values for each function.
