Curve sketching practice question 3
Which of the following functions is even?
What must equal for an even function? Test every piece of each choice, including the pieces with inside.
Even means . Replace by in each and simplify; remember cosine is even and sine is odd.
Theory UBC Math 100 §7.4CLP-1 §3.6Key results
Worked solution
Strategy
Test the definition directly. A sum of even functions is even; mixing an odd term in breaks the symmetry.
Working
Even powers are even, and cos is even, so cos(x³) is even although x³ is odd.
x4 + 5x² + sin(x³) adds an odd term to even ones; x³ sin x is even but the added x is odd; in (x² + 1)/(x − 1) the denominator is neither. None of those is even.
Answer
The answer is x to the fourth plus 5 x squared plus cosine of x cubed.
Check
At x = ±1, x4 + 5x² + cos(x³) gives 6 + cos 1 both times, while the version with sin(x³) gives 6 + sin 1 and 6 − sin 1.
Where this goes wrong
Ruling out cos(x³) because x³ is odd. What matters is the outer function: cosine of anything odd is still even, since cos(−u) = cos u.
The idea behind it
The curve sketching lesson, a figure to move, in a few steps
Leans on
Even and odd functions, from Functions, graphs and inverses
Try a similar question
Let . Find the domain of , show that is odd, and find a formula for .Saved in this browser only.