Odd/even functions; self-inverse practice questions.

Which of the following functions is odd?

• f(x) = x³ + 1
• f(x) = x³ − 4x
• f(x) = x² + x
• f(x) = |x|

Answer: f(x) = x³ − 4x

For f(x) = x³ − 4x, f(−x) = −x³ + 4x = −f(x), so it is odd. x³ + 1 fails because of the constant, x² + x is neither, and |x| is even.

Classify the function f(x) = x⁴ − 3x² + 7.

• odd
• even
• neither odd nor even
• both odd and even

Answer: even

All powers of x are even (the constant is x⁰), so f(−x) = f(x) for all x and the function is even.

The graph of an odd function is symmetric about which of the following?

• the y-axis
• the x-axis
• the line y = x
• the origin

Answer: the origin

f(−x) = −f(x) means the graph maps onto itself under a 180° rotation about the origin. Symmetry in the y-axis characterises even functions instead.

Let f and g both be odd functions. The product function fg is:

• even
• odd
• neither even nor odd
• impossible to determine

Answer: even

f(−x)g(−x) = (−f(x))(−g(x)) = f(x)g(x), so the product of two odd functions is always even — analogous to (−1) × (−1) = +1.

Which of the following functions is self-inverse on its domain?

• f(x) = x + 4
• f(x) = 4x
• f(x) = x⁴
• f(x) = 4/x

Answer: f(x) = 4/x

For f(x) = 4/x, f(f(x)) = 4/(4/x) = x, so f⁻¹ = f. The inverses of x + 4 and 4x are x − 4 and x/4, and x⁴ is not even one-to-one on the reals.

The function f(x) = (kx + 5)/(2x − 3), x ≠ 3/2, satisfies f(f(x)) = x for all x in its domain. Determine the value of k.

• k = −3
• k = 2
• k = 5
• k = 3

Answer: k = 3

With k = 3, f(f(x)) = (3f(x) + 5)/(2f(x) − 3) = (9x + 15 + 10x − 15)/(6x + 10 − 6x + 9) = 19x/19 = x. In general (ax + b)/(cx + d) is self-inverse when d = −a.