Questions and Answers
We exclude from a function's domain real numbers that cause division by ____.
zero
Real numbers that result in a square root of a _______ _______ are excluded from a function's domain.
negative number
The domain of f(x)=9x+5 consists of all real numbers, represented in interval notation as ____.
(-infinity, infinity)
The domain of g(x)=3 / (x+7) consists of all real numbers except -7, represented in interval notation as _______.
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The domain of h(x)=1/x + 3/x + 1 consists of all real numbers except 0 and -1, represented in interval notation as _______.
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The notation fog, called the _____ of the function f with g, is defined by _______.
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The function (fog) is found by replacing each occurrence of ___ in the equation for _____.
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The notation gof, called the _____ of the function g with f, is defined by ____.
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Fog is the same as function gof?
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F(g(x))=f(x)*g(x)
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If f(g(x))=3/g(x)-9 and g(x)=27/x, then 0 and ___ must be excluded from the domain of fog.
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What is the domain of the function: f(x)=14(x-10)?
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Find the domain of the function: f(x)=12/(x+8).
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Find the domain of the function: f(x)=48/(x^2+12x-63).
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Find the domain of the function: f(x)=5/(x^2+1) + 3/(x^2-16).
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Find the domain of the function: f(x)=sqrt(x-9).
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Find the domain of the function: f(x)=sqrt(x-4) + sqrt(x+13).
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Define the term 'radicand'.
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Study Notes
Domains of Functions
- Real numbers causing division by zero are excluded from a function's domain.
- Square roots of negative numbers are not included in the domain of a function.
- The domain of f(x) = 9x + 5 is all real numbers, expressed as (-infinity, infinity).
- For g(x) = 3 / (x + 7), exclude -7 from its domain: (-infinity, -7) U (-7, infinity).
- The domain of h(x) = 1/x + 3/(x + 1) excludes 0 and -1: (-infinity, -1) U (-1, 0) U (0, infinity).
Composite Functions
- Composition of functions f and g, denoted as fog, is defined by (fog)(x) = f(g(x)).
- To find fog, replace g(x) in the function f with g(x).
- Composition of functions g and f, denoted as gof, is defined by (gof)(x) = g(f(x)).
- fog and gof represent different functions; they are not the same.
Function Composition Truths
- The equation f(g(x)) does not equal f(x) * g(x).
- For f(g(x)) = 3/g(x) - 9 with g(x) = 27/x, exclude both 0 and 3 from the domain of fog.
Finding Domains of Specific Functions
- The function f(x) = 14(x - 10) has a domain of (-infinity, infinity).
- The function f(x) = 12/x + 8 restricts its domain to (-infinity, -8) U (-8, infinity).
- For f(x) = 48/(x^2 + 12x - 63), identify roots through factorization resulting in domain (-infinity, -9) U (-9, 7) U (7, infinity).
- In f(x) = 5/(x^2 + 1) + 3/(x^2 - 16), only x^2 - 16 causes division by zero, leading to domain (-infinity, -4) U (-4, 4) U (4, infinity).
Square Root Functions
- For functions involving square roots, only non-negative radicands are allowed.
- For f(x) = √(x - 9), the domain is [9, infinity) since x - 9 must be greater than or equal to zero.
- For f(x) = √(x - 4) + √(x + 13), impose conditions on both radicands: x - 4 ≥ 0 and x + 13 ≥ 0. This yields a domain of [4, infinity).
Key Terms
- Radicand refers to the number or expression inside a radical symbol.
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Description
Test your knowledge of combinations and composite functions in Algebra. This quiz covers essential concepts, including domains and exclusions for function values. Perfect for reinforcing your understanding of function behavior in mathematics.
