0. Functions

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x{}^5-x^3-2$

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

d. R(x) = √ 2x - 1

In Exercises 7–10, determine from its graph if the function is one-to-one.

f(x) = 3 - x, x < 0

= 3, x ≥ 0

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

𝔂 = sec x tan x

Identify the symmetry (if any) in the graphs of the following equations.

$y^2-4x^2=4$

In Exercises 7–10, determine from its graph if the function is one-to-one.

f(x) = 1 - x/2, x ≤ 0

x/(x + 2), x > 0

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=1$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.

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

State whether each function is increasing, decreasing, or neither.

d. Kinetic energy as a function of a particle’s velocity

State whether each function is increasing, decreasing, or neither.

c. Height above Earth’s sea level as a function of atmospheric pressure (assumed nonzero)

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 = (−x)²/³

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

𝔂 = e⁻ˣ²

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$\text{ƒ}\left(x\right)=x\mid x\mid$

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

𝔂 = x cos x