Explicit, implicit, parametric; even and odd: how a function is written and how it looks

The problem

Fold a paper along a line and cut a shape: open it, and both halves match. The graph of y = x² does exactly that about the y-axis. The graph of y = x³ does something else: turn it half a circle about the origin and it lands on itself.

This section is about how functions are written, explicitly, implicitly or with a parameter, and about these two symmetries, even and odd.

A function f is even if f(−x) = f(x), and odd if f(−x) = −f(x), for every x in its domain; −x must also be in the domain. Many functions are neither.

Key terms

Explicit function

y is expressed directly in terms of the independent variable x.

Implicit function

x and y are so mixed up that y is not expressed in terms of x. It is written f(x, y) = 0.

Parametric equations

Both x and y are given in terms of a third variable, a parameter t or θ. Eliminating the parameter gives the curve's ordinary equation.

Even and odd

Even: f(−x) = f(x), e.g. x² and cos x; the graph mirrors across the y-axis. Odd: f(−x) = −f(x), e.g. x³ and sin x; the graph matches itself after a half-turn about the origin.

Short questions with model answers

1. Q1. Show that the parametric equations x = a cos t and y = a sin t represent the circle x² + y² = a².

   • x² = a² cos²t, y² = a² sin²t
   • x² + y² = a²(cos²t + sin²t) = a²

   x² + y² = a²

2. Q2. Is f(x) = 3x/(x² + 1) even, odd or neither?

   • f(−x) = 3(−x)/((−x)² + 1) = −3x/(x² + 1)
   • f(−x) = −f(x)

   Odd

3. Q3. Is f(x) = sin x + cos x even, odd or neither?

   • f(−x) = sin(−x) + cos(−x) = −sin x + cos x
   • −sin x + cos x ≠ f(x) and ≠ −f(x)

   Neither even nor odd

Common mistakes

• ✗ Concluding that a function is odd just because it is not even.

  ✓ There is a third answer: neither. sin x + cos x is the book's example.

• ✗ Calling f(x) = (x + 2)² even because it is a square.

  ✓ f(−x) = (−x + 2)² = (x − 2)², which is not (x + 2)². Always replace x by −x inside the whole expression.

• ✗ Checking one value, finding f(−1) = f(1), and declaring the function even.

  ✓ Even and odd are about every x. One value can disprove symmetry, but proving it needs f(−x) worked out in general.

MCQs

1. 1. x² + xy + y² = 2 defines y as:

   1. (a) an explicit function
   2. (b) an implicit function
   3. (c) a parametric function
   4. (d) a constant function
   Show answer

   (b) x and y are mixed and y is not expressed in terms of x: an implicit function, f(x, y) = 0.

2. 2. f(x) = x³ + x is:

   1. (a) even
   2. (b) odd
   3. (c) neither
   4. (d) both
   Show answer

   (b) f(−x) = −x³ − x = −(x³ + x) = −f(x).

3. 3. f(x) = 3x⁴ − 2x² + 7 is:

   1. (a) even
   2. (b) odd
   3. (c) neither
   4. (d) not a function
   Show answer

   (a) Only even powers of x appear, so f(−x) = 3x⁴ − 2x² + 7 = f(x).

4. 4. The parametric equations x = at², y = 2at represent:

   1. (a) the circle x² + y² = a²
   2. (b) the parabola y² = 4ax
   3. (c) the ellipse x²/a² + y²/b² = 1
   4. (d) a straight line
   Show answer

   (b) t = y/2a, so x = a(y/2a)² = y²/4a, i.e. y² = 4ax.

5. 5. f(x) = (x + 2)² is:

   1. (a) even
   2. (b) odd
   3. (c) neither
   4. (d) both
   Show answer

   (c) f(−x) = (x − 2)², which is neither f(x) nor −f(x).

Quick revision

• Explicit: y = f(x). Implicit: f(x, y) = 0. Parametric: x = f(t), y = g(t); eliminate t to get the curve.
• Even: f(−x) = f(x), mirror across the y-axis. Odd: f(−x) = −f(x), half-turn about the origin.
• Many functions are neither; one value can disprove symmetry, but only the general f(−x) proves it.