Which Of The Following Is An Even Function

Even Functions: Definition, Identification, and Real‑World Examples

An even function is a mathematical function that exhibits perfect symmetry about the vertical axis. In practice, this property means that reflecting the graph of the function across the y‑axis leaves the graph unchanged. In algebraic terms, a function f satisfies the condition f(‑x) = f(x) for every value of x in its domain. Understanding even functions is fundamental in calculus, Fourier analysis, and many applied fields because the symmetry simplifies computations and reveals underlying patterns.

What Is an Even Function?

The defining characteristic of an even function is the equality of the function’s values at opposite points. Formally, a function f defined on a set D (where D is symmetric about zero) is even if:

• For all x ∈ D, f(‑x) = f(x).

This relationship can be visualized: if you draw a vertical line (the y‑axis) and mirror the graph, the mirrored image coincides exactly with the original. Classic examples include the parabola f(x) = x², the cosine wave f(x) = cos x, and the absolute‑value function f(x) = |x|. Each of these graphs is symmetric about the y‑axis.

How to Test Whether a Function Is Even

Identifying an even function involves a straightforward algebraic test:

1. Replace x with –x in the function’s expression.
2. Simplify the resulting expression.
3. Compare the new expression with the original.
   • If they are identical (or differ only by a constant factor that cancels out), the function is even.
   • If they are not identical, the function is not even (it may be odd, neither, or both).

Example:
Test f(x) = 3x⁴ – 2x² + 5.

• Compute f(‑x) = 3(‑x)⁴ – 2(‑x)² + 5 = 3x⁴ – 2x² + 5.
• Since f(‑x) = f(x), the function is even.

Common Even Functions and Why They Are Even

Below is a curated list of frequently encountered even functions, each illustrating the symmetry principle:

• Constant function: f(x) = c (any real number c). Because f(‑x) = c = f(x), constants are trivially even.
• Quadratic monomials: f(x) = x², f(x) = ax² + bx + c when b = 0. The term x² ensures symmetry.
• Higher even powers: f(x) = x⁴, f(x) = x⁶, etc. Even powers preserve the sign of x.
• Absolute value: f(x) = |x|. The absolute value discards sign, yielding symmetry.
• Cosine: f(x) = cos x. Cosine’s periodic nature repeats every 2π and is symmetric about the y‑axis.
• Secant: f(x) = sec x (where defined). Since sec(‑x) = sec x, it is even.
• Even combinations: f(x) = x² + sin² x. The sum of even components remains even.

Identifying Even Functions from a Given Set

Often, a question will present several functions and ask which are even. Consider the following set:

1. f₁(x) = x³ – x
2. f₂(x) = x⁴ + 2x²
3. f₃(x) = sin x
4. f₄(x) = |x| + x²
5. f₅(x) = eˣ

Step‑by‑step analysis:

1. f₁(x) = x³ – x

   • f₁(‑x) = (‑x)³ – (‑x) = –x³ + x = –(x³ – x) = –f₁(x).
   • This is an odd function, not even.

2. f₂(x) = x⁴ + 2x²

   • f₂(‑x) = (‑x)⁴ + 2(‑x)² = x⁴ + 2x² = f₂(x).
   • Even function.

3. f₃(x) = sin x

   • f₃(‑x) = sin(‑x) = –sin x = –f₃(x).
   • Odd function.

4. f₄(x) = |x| + x²

   • f₄(‑x) = |‑x| + (‑x)² = |x| + x² = f₄(x).
   • Even function.

5. f₅(x) = eˣ

   • f₅(‑x) = e^(‑x) = 1/eˣ ≠ eˣ (except at x = 0).
   • Neither even nor odd.

Result: The even functions in this collection are f₂(x) and f₄(x).

Graphical Interpretation

A visual check reinforces the algebraic test:

• Plot f₂(x) = x⁴ + 2x². The curve rises symmetrically on both sides of the y‑axis.
• Plot f₄(x) = |x| + x². The V‑shape of |x| merges with the parabola, yet the overall graph remains symmetric.

If a graph looks the same after a mirror reflection across the y‑axis, the underlying function is even.

Scientific Explanation: Why Even Functions Matter

Even functions are more than a textbook curiosity; they play key roles in advanced mathematics and engineering:

• Fourier Series: Any periodic function can be expressed as a sum of sines (odd) and cosines (even). The cosine terms capture the even component of the signal.
• Integration: When integrating an even function over a symmetric interval ([-a, a]), the integral simplifies to twice the integral from 0 to a:
  [ \int_{-a}^{a} f(x