Question

Which of the ten basic functions in

our toolkit have graphs that look the
same when reflected of the y-axis
and then reflected over the x-axis?

Answer 1

a) x³, 1/x, x, sin(x)

Explanation:

You want to know which list of functions has the same graph when reflected across both the y- and x-axes.

Odd function

Reflection across the y-axis transforms the function to ...

f(x) ⇒ f(-x)

Reflection across the x-axis transforms the function to ...

f(x) ⇒ -f(x)

Hence, reflection across both axes transforms the function to ...

f(x) ⇒ -f(-x)

A function that has the characteristic that f(x) = -f(-x) is an odd function. Its graph is symmetrical about the origin. We can check to see if the functions of interest match this definition of an odd function:

x³ = -(-x)³ . . . . true

1/x = -(1/-x) . . . . true

x = -(-x) . . . . true

sin(x) = -sin(-x) . . . . true

cos(x) = -cos(-x) . . . . false

e^x = -e^-x . . . . false

|x| = -|-x| . . . . false

The basic functions that are odd are ...

x³, 1/x, x, sin(x)