10.6. Solved problems

1. Give the intervals for which the function: f(x) = |x - 3|, is increasing and decreasing.

   Decreasing in:  (-∞, 3]
   Increasing in:  [3, +∞)
   

2. Determine whether the function: f(x) = 2x³ - x, is odd or even.

   f(-x) = 2(-x)3 - (-x) = -2x3 + x = -(2x3 - x) = -f(x)
   

   Therefore, f(x) is an odd function.

3. Is equation y = 3x + 2 a function of x or not?

   For each value of x, there is only one possible value of y; therefore, thr right hand side of the equation defines a function.

4. Is equation: y = x² + 3x + 2 a function x or not?

   For each value of x, there is only one possible value of y; therefore, right hand side defines a function.

5. Is equation: y² - 2x - 1 = 0 a function of x or not?

   Solving for y, we get:

         ______
   y = ±√2x + 1
   

   For each value of x, there is may be more than one value of y; therefore, equation right hand side is not a function of x.

6. Is equation: x² + y² - 4 = 0 a function of x or not?

   Solving for y, we get:

          ______
    y = ±√4 - x2
   

   For each value of x, there is may be more than one value of y; therefore, equation is not a function of x.

7. Is list of ordered pairs a function or not? (0, 6), (1, 5), (2, 4), (3, 3), (4, 2)

   For each value of x, there is only one value of y; therefore, list defines a function.

8. List ordered pairs for integer values of x : 0...9 for the function:

                  ____
   f(x) = floor( √ x3 )
   
   x    f(x)
   ----------
   0      0
   1      1
   2      2
   3      5
   4      8
   5     11
   6     14
   7     18
   8     22
   9     27
   

9. List ordered pairs for integer values of x : 0...9 for:

                 _______
   g(x) = ceil( √ x + 4  )
   
   x    f(x)
   ----------
   0      2
   1      3
   2      3
   3      3
   4      3
   5      3
   6      4
   7      4
   8      4
   9      4
   

10. Determine the domain and range of: d(x) = x² - 4

    Domain is the set of real numbers: (-∞, +∞)
    Range is the set: [-4, +∞)

11. Determine the domain of: f(x) = (x + 2)/(x - 2)

    All real numbers; x ≠ 2

12. Determine the domain of: g(x) = 1/(x² - 4)

    All real numbers; x ≠ 2, x ≠ -2

13. Determine the domain and range of: h(x) = 2^x

    Domain is the set of real numbers: (-∞, +∞) Range is the set: (0, +∞)

14. Find the value of k that makes the function continuous:

            2x + k   if x ≤ 1  
    f(x) = 
            x2 + 1   if x > 1
    
    Need to solve 2x + k = x2 + 1 at the boundary, x = 1:
    
    2 + k = 2
    k = 0
    

15. Graph f(t) = u(t - 1) - u(t - 3) (difference of unit step functions)

    

16. Graph f(t) = 1 - 2^-t for t ≥ 0

    

17. Define a postal function, the cost of mailing a US first class letter by weight from 0 to 9 ounces.

    
           $0.37    if 0 < x ≤ 1oz
           $0.60    if 1 < x ≤ 2oz
           $0.83    if 2 < x ≤ 3oz
           $1.06    if 3 < x ≤ 4oz
    p(x) = $1.29    if 4 < x ≤ 5oz
           $1.52    if 5 < x ≤ 6oz
           $1.75    if 6 < x ≤ 7oz
           $1.98    if 7 < x ≤ 8oz
           $2.21    if 8 < x ≤ 9oz
    

18. Evaluate: f(x) = x² + 3x + 2 for x = c + 1

    f(x) = x2 + 3x + 2
    
         = (c + 1)2 + 3(c + 1) + 2
    
         = c2 + 2c + 1 + 3c + 3 + 2
    
         = c2 + 5c + 6
    

19. Evaluate: g(x) = [f(x + h) - f(t)]/h for f(x) = x² - 2x + 1

           (x + h)2 - 2(x + h) + 1 - x2 + 2x - 1
    g(x) = -------------------------------------
                        h
    
           x2 + 2xh + h2 - 2x - 2h + 1 - x2 + 2x - 1
    g(x) = -----------------------------------------
                             h
    
            2xh + h2 - 2h
    g(x) = --------------- = 2x - 2 + h
                  h
    

20. For f(x) = 2^x and g(x) = 3^x evaluate f(2)g(2), f(g(2)), and g(f(2)).

    f(2)g(2) = 22×32 = 4×9 = 36
    
    f(g(2))  = 29 = 512
    
    g(f(2))  = 34 = 81
    

21. For f(x) = x^-1 and g(x) = x³ - 1 find f(c)g(c), f(g(c)), and g(f(c)).

    f(c)g(c) = c-1·(c3 - 1) = (c3 - 1)/c
    
    f(g(c))  = (c3 - 1)-1 = 1/(c3 - 1) 
    
    g(f(c))  = (c-1)3 - 1 = (1/c3) - 1
    

22. For f(x) = x² and g(x) = √x find f(x)g(x), f(g(x)), and g(f(x)).

                   _
    f(x)g(x) = x2·√x
                  _
    f(g(x))  = ( √x )2 = x
                ___
    g(f(x))  = √ x2 = x
    

23. Does the graph of f(x) = x² - x - 2 cross the x-axis in the range [0, 4]?

    f(0) = 02 - 0 - 2 = -2 
    f(4) = 42 - 4 - 2 = 10
    

    Since the value of f(x) has opposite signs at the endpoints of the interval; the graph of f(x) must cross the x-axis.