High School Common Core Mathematics

Front 1

If f is a function and x is an element of its domain, then f(x) denotes what?

Back 1

f(x) denotes the output of f corresponding to the input x.

Front 2

If f is a function, the graph of f is the graph of what equation?

Back 2

The graph of f is the graph of the equation y=f(x)

Front 3

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.

Back 3

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.

Front 4

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?

Back 4

C(x)/x=1.25+2500/x

Front 5

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)

Back 5

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

Front 6

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

Back 6

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.

Front 7

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

Back 7

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

Front 8

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

Back 8

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

Front 9

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.

Back 9

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.)

Front 10

Sequences are functions, sometimes defined recursively, whose domain is what?

Back 10

Their domain is a subset of the integers.

Front 11

What is an example of a sequence that is a function, defined recursively?

Back 11

The Fibonacci sequence is a sequence that is a function which is defined recursively.

Front 12

How is the Fibonacci sequence defined in function notation?

Back 12

f(0) = f(1) = 1
f(n+1) = f(n)+f(n-1) for n≥1