1. Which of these graphs of relations describe y as a function of x?
That is, which are graphs of functions? Answer(s): ____________
(no explanation required.) (There may be more than one graph which represents a function.)
2. Consider the points (6, -5) and (2, 3).
(a) State the midpoint of the line segment with the given endpoints. (No work required)
(b) If the point you found in (a) is the center of a circle, and the other two points are points on the circle, find the length of the radius of the circle. (That is, find the distance between the center point and a point on the circle.) Find the exact answer and simplify as much as possible. Show work.
3. Consider the following graph of y = f(x).
(no explanations required)
(a) State the x-intercept(s).
(b) State the y-intercept(s).
(c) State the domain.
(d) State the range.
4. Let f(x) = (x - 6)/(x2- 1)
(a) Calculatef(-2). (work optional)
(b) State the domain of the function f(x) = (x - 6)/(x2- 1)
(c) Find f (a+1) and simplify as much as possible. Show work.
5. f is a function that takes a real number x and performs these three steps in the order given:
(1) Take the absolute value.
(2) Subtract 4 from the result.
(3) Take the reciprocal of the result.
(That is, make the quantity the denominator of a fraction with numerator 1.)
(a) Find an expression for f(x). (no explanation required)
(b) State the domain of f. (no explanation required)
6. Given f(x)=√(1-x) and g(x)=x, which of the following is the domain of the quotient function ? Explain. 6._______
A. (-∞,0)
B. (-∞,0)∪(0,∞)
C. (-∞,0)∪(0,1)
D. (-∞, 1]
7. For income x (in dollars), a particular state's income tax T (in dollars) is given by
(a) What is the tax on an income of $6,500? Show some work.
(b) What is the tax on an income of $65,000? Show some work.
8. Let y = 4 - x2.
(a) Find the y-intercept(s) of the graph of the equation, if any exist. (work optional)
(b) Find the x-intercept(s) of the graph of the equation, if any exist. (work optional)
(c) Create a table of sample points on the graph of the equation (include at least six points), and use them to help create a graph of the equation.(You may use the grid shown below, hand-draw and scan, or you may use the free Desmos graphing calculator described under Course Resource to generate a graph, save as a jpg and attach.)
(d) Is the graph symmetric with respect to the x-axis? _____ (yes or no).If no, state a point on the graph and state the appropriate reflection point which fails to be on the graph.
(e) Is the graph symmetric with respect to the y-axis? _____ (yes or no).If no, state a point on the graph and state the appropriate reflection point which fails to be on the graph.
(f) Is the graph symmetric with respect to the origin? _____ (yes or no).If no, state a point on the graph and state the appropriate reflection point which fails to be on the graph.
9. Let f(x) = 3x2+7x-5 and g(x) = 2 - x.
(a) Evaluate the function g - f for x = -1. That is, find (g - f) (-1). Show work.
(b) Evaluate the function fg for x = -1. That is, find (fg) (-1). Show work.
(c) Find the difference function (f - g) (x) and simplify the results. Show work.
10.The cost, in dollars, for a company to produce x widgets is given by C(x) = 3600 + 5x for x ≥ 0, and the price-demand function, in dollars per widget, is p(x) = 45 - 0.04x for 0 ≤ x ≤ 1125.
(a) Find and interpret C (200).
(b) Find and interpret C ¯(200). (Note that C ¯(x) is the average cost function.)
(c) Find and simplify the expression for the revenue function R(x). (Work optional)
(d) Find and simplify the expression for the profit function P(x). (Work optional)Note that p(x) and P(x) are different functions.
(e) Find and interpret P (200), where P(x) is the profit function in part (d).
11. Determine whether or not the equation represents y as a function of x. Show work
x2 = y2
12. According to this website, the census data for Lorain County, Ohio is: Show Work
|
Year
|
1970
|
1980
|
1990
|
2000
|
|
Population
|
256843
|
274909
|
271126
|
284664
|
Find the least squares regression line for these data and comment on the goodness of fit. Interpret the slope of the line of best fit.
Use the regression line to predict the population of Lorain County in 2010. (The recorded figure from the 2010 census is 301,356)
(c) Use the regression line to predict when the population of Lake County will reach 325,000.