Sample Exercises  
Chapter 3: New Functions from Old
 
  1. Suppose c represents the function where x is a certain state college and c(x) is the cost of tuition for the college in hundreds of dollars. Match the following situations to the formulas given below. (Not all the functions will be used.)

    2.  
    (i)   y = 0.1c(x)  (ii)   y = 0.9c(x)  (iii)   y = c(x) + 1
    (iv)   y = 1.1c(x) (v)   y = -0.1c(x)  (vi)   y = c(x) - 1 
    (vii)   y = c(x) - 10  (viii)   y = c(x) + 10 (iv)   y = c(x) + 0.1 
     
    1. All tuition costs are increased by 100 dollars.
    2. All tuition costs are decreased by 100 dollars.
    3. All tuition costs are increased by 10%.
    4. All tuition costs are decreased by 10%.
     
  2. Answer the following questions about transformations of f with the constant a.
    1. Suppose f(1) > 0 and 0 < af(1) < f(1). What do you know about a?
    2. Suppose f(1) > 0 and af(1) > f(1). What do you know about a?
    3. Suppose f(1) > 0 and a + f(1) > f(1). What do you know about a?
    4. Suppose f(1) > 0 and a + f(1) < f(1). What do you know about a?
     
  1. The following is a graph of f.
    Sketch a graph of each of the following.
     
    1. y = f(x) + 2
    2. y = 2f(x) + 1
    3. y = -2f(x) - 3
     
  1. Suppose a population of a town is growing at a rate of 3% per year. If the population in 1970 was 2000 the function describing the growth can be modeled by p(t) = 2000(1.03)t where t is the time since 1970. Which of the following situations describes a vertical stretch by a factor of 4?

    2. A. The population of a town in 1970 was 8000 and has increased by 3% per year.
    B. The population of a town in 1970 was 2000 and has increased by 12% per year.

     

  2. In the United States, the function that converts the length of someone's foot to a man's shoe size is different than the one that converts foot length to a woman's shoe size. The table below shows the functions that convert the length of a person's foot, in inches, to a man's shoe size and a woman's shoe size.
 
Foot Length (inches) 8 9 10 11 12
Men's Shoe Size 2 5 8 11 14
Women's Shoe Size 3.5 6.5 9.5 12.5 15.5
 
    1. Let m represent the function where the input, x, is the length of a person's foot and the output, m(x), is the men's shoe size that person would wear. What kind of a function is this?
    2. Determine the formula where the input, x, is the length of a person's foot and the output, m(x), is the men's shoe size that person would wear.
    3. Let w represent the function where the input, x, is the length of a person's foot and the output, w(x), is the women's shoe size that person would wear. What is the relationship between w(x) and m(x)? Use this relationship to determine a formula for w.
  1. What is the difference between algebraically combining two functions and composing two functions?
  1. Let f and g be given by the following table. Where possible, fill in the third row giving the outputs for g(f(x)).
 
x
-3
-2
-1
0
1
2
3
f(x)
4
-1
0
2
3
-2
1
g(x)
-2
0
2
4
2
1
-1
g(f(x))              
 
  1. You may have heard that it is possible to estimate the temperature by counting the number of cricket chirps. According to Insect Fact and Folklore the temperature in degrees Fahrenheit is approximately equal to the number of chirps in fifteen seconds plus 38.
    1. Find the equation for the function whose input is number of chirps in fifteen seconds and whose output is temperature in degrees Fahrenheit.
    2. Your answer to part (a) should be a linear function. Explain the physical meaning of the slope and y-intercept.
    3. Suppose you wanted the temperature in degrees Celsius rather than degrees Fahrenheit. Explain why this will be a composition.
    4. Find the equation for the function whose input is number of chirps and whose output is temperature in degrees Celsius. [Note: C = (5/9)(F - 32).]
    5. Your answer to part (d) should be a linear function. Explain the physical meaning of the slope and the y-intercept.
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