(1)

A function f is said to be odd provided f(-x) = - f(x) for all x in the domain of f. Which of the following functions are odd?

(d) u(y) = y4

(e) w(a) = a-3/5

(f) q(r) = r3 + r - 1

(2)

A function f is said to be even provided f(-x) = f(x) for all x in the domain of f. Which of the following functions are even?

(a) f(x) = x5

(b) g(t) = t4 - 3t + 2

(c) h(z) = z4 - z2 + 10

(d) u(y) = [Ö((y2 + 1))]

(e) w(a) = 2 + a-4/3

(f) q(r) = 2

(3)

Choose any of the odd functions above. If you translate this function one unit left, is the resulting function still odd?

(4)

Choose any of the (nonconstant) even functions above. If you translate this function one unit left, is the resulting function still even?

(5)

If we take an even function and translate it vertically one unit, is the resulting function still even? Justify your answer.

(6)

If we take an odd function and translate it vertically one unit, is the resulting function still odd? Justify your answer.

(7)

Explain why the function y = 0 is both even and odd. Is this true for any other constant function?

(8)

Explain why an even function cannot be invertible.

(9)

Suppose that f is an odd invertible function. Is the inverse of f also odd? Justify your answer.

(10)

Let f(x) = x² - 3x + 1. Find a formula for h(x) = f(x+1) +3. Without sketching either the graph of f or h, what can we say about how these graphs compare?

(11)

Find a formula for the function h which is obtained from f(x) = Öx by first scaling by a factor of -1/2, then translating 5 units down and 4 units left.

     

                                      Figure 1                                                                          Figure 2

(12)

The graph in Figure 1 above is a scaled and translated version of f(x) = [1/x]. Find a formula for this function.

(13)

The graph in Figure 2 above is a scaled and translated version of f(x) = x³. Find a formula for this function.

(14)

If f(x) = x³ - 2, find a formula for f ^-1 as a function of the output variable y.

(15)

If g(t) = 5(x-2)^1/5, find a formula for g ^-1 as a function of the output variable y.

(16)

Graphs of functions f and g are given below. The symbol f°g stands for the composition y = f(g(x)).  On the grid provided, sketch the graphs of f°g and g°f.

(17)

Let f(x) = x² - x + 1 and let g(u) = u + 5. Find formulas for f°g and g°f.

(18)

Let f(t) = [Ö(t - 1)] and let g(z) = [1/z]. Find formulas for f°g and g°f.

(19)

Show that f(x) = x^1/5 - 3 and g(x) = (x+3)⁵ are inverses of each other.

(20)

Show that f(x) = 8(x-1)³ and g(x) = [(x^1/3)/2] + 1 are inverses of each other.

(21)

On the grids provided in Figures 3 and 4 below, reflect the graphs of f and g about the line y = x. Explain why the resulting graphs prove f has and inverse while g does not.

                                Figure 3                                                                                     Figure 4

(22)

The graph of a function f is shown below. On the same grid, sketch the graph of f ^-1.

(23)

Let f and g be functions. In your own words, describe the difference between the functions fg and f°g.

(24)

Let f(x) = Log(x) and g(x) = x² - 1. Find formulas for f+g, g/f, f/g, fg, and f°g.