- Topic
- Curve sketching
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Curve sketching practice question 5
Which of the following functions is odd, so that its graph is symmetric about the origin?
In symbols, what does symmetry about the origin say about ? Test each choice against that, factor by factor and term by term.
Odd means for every . Replace by in each and simplify, remembering that and are odd and is even.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Test the definition: replace x by −x in the whole formula and compare the result with −f(x).
Working
An even factor times an odd factor is odd.
x sin x is even; the other two are neither, because each mixes an odd part with an even one.
Answer
The answer is x squared times tan x.
Check
x² tan x is π²/16 at x = π/4 and −π²/16 at x = −π/4, opposite values as an odd function needs, while x sin x is π√2/8 at both.
Where this goes wrong
Calling x sin x odd because both factors are odd. As with two negative numbers, the two sign changes cancel: the product of two odd functions is even.
Leans on
Even and odd functions, from Functions, graphs and inverses
Try a similar question
Which of the following functions is even?Saved in this browser only.