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12 Cards in this Set
- Front
- Back
If f is a function and x is an element of its domain, then f(x) denotes what?
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f(x) denotes the output of f corresponding to the input x.
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If f is a function, the graph of f is the graph of what equation?
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The graph of f is the graph of the equation y=f(x)
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Katy is told that the cost of producing x DVDs is given by C(x)=1.25x+2500. She is then asked to find an equation for C(x)/x, the average cost per DVD of producing x DVDs. She works through simplifying the equation as such
C(x)/x=(1.25x+2500)/x C=1.25+2500/x Is Katy's equation for finding the average cost per DVD of producing x DVDs correct? Explain. |
Katy's equation is not correct. She has interpreted function notation C(x) as multiplication notation. C(x) is a notation for the number that C assigns to x, not the result of multiplying C and x.
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Katy is told that the cost of producing x DVDs is given by C(x)=1.25x+2500. She is then asked to find an equation for C(x)/x, the average cost per DVD of producing x DVDs. What is the correct equation to find the average cost?
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C(x)/x=1.25+2500/x
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List the algebraic operations in order of evaluation. What restrictions does each operation place on the domain of the function? What is the function's domain? y=2/(x-3)
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Operation: subtract 3 from x. This does not restrict the domain, since we can subtract 3 from any number.
Operation: divide 2 from the above result. This means that x-3 can't equal 0, so x≠3. The domain is x≠3 |
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List the algebraic operations in order of evaluation. What restrictions does each operation place on the domain of the function? What is the function's domain? y=(√x-5)+1
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Operation: subtract 5 from x. This does not restrict the domain, since we can subtract 5 from any number.
Operation: take the square root of the above result. This means that x-5 can't be negative so x≥5. Operation: Add 1 to the above result. This does not restrict the domain, since we can add 1 to any number. The domain is x≥5. |
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List the algebraic operations in order of evaluation. What restrictions does each operation place on the domain of the function? What is the function's domain? y=7/4-(x-3)2
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Operation: subtract 3 from x. This does not restrict the domain, since we can subtract 3 from any number.
Operation: square the result. This does not restrict the domain, since we can square any number. Operation: subtract the above result from 4. This does not restrict the domain, since we can subtract any number from 4. Operation: divide 7 by the above result. This means that the result of the above step can't equal 0, so (x-3)2≠4. Therefore, x-3≠2 x≠5 and x-3≠-2 x≠1 The domain is all x except x=1,5 |
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List the algebraic operations in order of evaluation. What restrictions does each operation place on the domain of the function? What is the function's domain? y=4-(x-3)^½
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Operation: subtract 3 from x. This operation does not restrict the domain, since we can subtract 3 from any number.
Operation: raise the above result to the power of 1/2. This means that x-3 can't be negative, so x≥3 Operation: subtract the above result from 4. This operation does not restrict the domain, since we can subtract any number from 4. The domain is x≥3 |
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Given a function f, is the statement f(x+h)=f(x)+f(h) true for any two numbers x and h? If so, prove it. If not, find a function for which the statement is true and provide an example of a function for which the statement does not hold true.
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The statement does not hold true for all functions.
A function for which it holds is the function f given by f(a)=5a. If f(a)=5a, then f(x+h)=5(x+h)=5x+5h= f(x) +f(h). A function for which it does not hold true is the function f given by f(a)=a². If f(a)=a², then f(x+h) = (x+h)²= x²+2xh+h², which is different than f(x)+f(h) = x²+h². (NOTE: Student's examples may differ. There are many valid examples of functions that do and do not hold true for the given statement.) |
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Sequences are functions, sometimes defined recursively, whose domain is what?
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Their domain is a subset of the integers.
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What is an example of a sequence that is a function, defined recursively?
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The Fibonacci sequence is a sequence that is a function which is defined recursively.
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How is the Fibonacci sequence defined in function notation?
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f(0) = f(1) = 1
f(n+1) = f(n)+f(n-1) for n≥1 |