0. Functions

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x⁵ - x³ - x

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = 1/|x|

State whether the functions represented by graphs A , B , C and in the figure are even, odd, or neither. <IMAGE>

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = 1 - cos x

Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?

17. Even-odd decompositions

b. If f(x) = f_E(x) + f_O(x) is the sum of an even function f_E(x) and an odd function f_O(x), then show that

f_E(x) = (f(x)+f(-x))/2 and f_O(x) = (f(x)-f(-x))/2

Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. <IMAGE>

a. At which points is the slope of the curve negative?

Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).

Suppose that ƒ and g are both odd functions defined on the entire real line. Which of the following (where defined) are even? odd?

a. ƒg

b. ƒ³

c. ƒ(sin x)

d. g(sec x)

e. |g|

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x - sin x

Given the graph of the following function, determine the intervals on which $f\left(x\right)$  is decreasing.

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x²/⁵

Find the largest interval on which the given function is increasing.

c. g(x) = (3x - 1)¹/³

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = x³/8

In Exercises 9–16, determine whether the function is even, odd, or neither.

𝔂 = x² + 1