Functions
Transformations and Piecewise-defined Functions
Function Analysis
Quadratic Functions
Polynomial Functions
100
Find the domain of the function and write it in interval notation:
f(x)=sqrt(x+5)
What is
x in [-5, oo)?
100
The graph of the function y=f(x) has a point with coordinates (4,3). What is the corresponding point on the transformed graph g(x)=-3f(x)?
What is (4, -9)?
100
Find the domain and the range from the graph of the function.
What is
x in [-2,2]
and
y in [-2,2]
?
100
Find the vertex of
f(x)=x^2+8x+10
What is (-4, -6)?
100
The polynomial function
f(x)=0.25(x+1)(x-4)^2(x-2)
is given. Determine the degree of the polynomial, its leading coefficient, and the end behavior.
What is n=4, a=0.25, up on the far left (
x->-oo, f(x)->oo
) and up on the far right (
x->oo, f(x)->oo
)?
200
Let
f(x)=5x-3
and
g(x)=4x^2-1
Find the function
(f+g)(x)
and write its domain in interval notation.
What is
(f+g)(x)=4x^2+5x-4
and the domain
x in (-oo, oo)
?
200
The graph of the function y=f(x) has a point with coordinates (4,3). What is the corresponding point on the transformed graph g(x)=f(x-4)-7?
What is (8,-4)?
200
List the intervals on which f(x) is increasing and where it is decreasing.
What is decreasing on
x in (-2,-1) uu (1,2)
and increasing on
x in (-1,1)
?
200
Find the axis of symmetry of
f(x)=x^2+8x+10
What is x=-4?
200
The polynomial function
f(x)=0.25(x+1)(x-4)^2(x-2)
is given. Find all zeros and list their multiplicity.
What is x=-1, multiplicity 1; x=4, multiplicity 2; and x=2, multiplicity 1?
300
Let
f(x)=5x-3
and
g(x)=4x^2-1
Find the function
(f/g)(x)
and write its domain in interval notation.
What is
(f/g)(x)=(5x-3)/(4x^2-1)
and domain
x in (-oo,-1/2) uu (-1/2,1/2) uu (1/2,oo)
?
300
The graph of the function y=f(x) has a point with coordinates (4,3). What is the corresponding point on the transformed graph g(x)=-3f(x+2)+1?
What is (2, -8)?
300
Is the function odd, even, or neither?
What is odd?
300
Rewrite
f(x)=x^2+8x+10
in standard (vertex) form.
What is
f(x)=(x+4)^2-6
?
300
The polynomial function
f(x)=0.25(x+1)(x-4)^2(x-2)
is given. Find the y-intercept.
What is (0,-8)?
400
Let
f(x)=5x-3
and
g(x)=4x^2-1
Find their composition,
f(g(x))
What is
f(g(x))=20x^2-8
?
400
Describe the transformations of
y=|x|
and graph the function
f(x)=-3|x+2|-1
What is reflect across the x-axis, stretch by a factor of 3, shift 2 units to the left and one unit down
?
400
Does the function have any local minimum or maximum?
Does the function have absolute minimum or maximum?
What is (-1, -2) is a local and an absolute minimum, (1,2) is a local and absolute maximum?
400
Find the x-intercepts and y-intercept of
f(x)=x^2+8x+10
What is
(-4-sqrt(6), 0) and (-4+sqrt(6), 0)
and (0,10)?
400
An open box is to be constructed from a rectangular sheet of metal, 8 inches wide and 15 inches long, by cutting out squares with sides of length x from each corner and bending the sides. Express the volume V as a function of x.
Find the domain to account for the physical restrictions on the values of x. Explain your reasoning.
What is V=4x(4-x)(7.5-x)?
The height has to be less than 4. No side can be less than or equal to 0.
500
Let
H(x)=sqrt(x+5)
Express it as a composition of two functions; that is find two functions f(x) and g(x) such that
H(x)=f(g(x))
What is
f(x)=sqrt(x)
and
g(x)=x+5
?
500
A piecewise-defined function
F(x)={(x^2,-3<x<0),(2x-5, 0<=x):}
is given. Find F(-2), F(-1), F(0), F(1), F(2).
What is F(-2)=4, F(-1)=1, F(0)=-5, F(1)=-3, F(2)=-1?
500
Let
f(x)=2x^3-8
Find the average rate of change of f(x) from x=-1 to x=3.
What is 14?
500
A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides. Given 100 feet of wire fencing, determine the dimensions that would create a garden with maximum area. What is the maximum area?
What is width is 25 ft, length is 50 ft, area is 1,250 square feet?
500
The polynomial function
p(x)=2x^4+5x^3+5x^2+20x-12
is given. Use synthetic division to check that x=1/2 is a rational zero of p(x). Factor polynomial completely over the set of complex numbers.
What is
p(x)=2(x-1/2)(x+3)(x-2i)(x+2i)
?