Funcitons

Bijective Function

A function that is both injective and surjective.

Injective Function (One-to-One Function)

For \(x_1, x_2\) in the domain of \(y = f(x)\), if \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\).

• \(y = x + 1\), \(x \in \mathbb{R}\) is an injective function.
• \(y = \sin(x)\), \(x \in \mathbb{R}\) is not an injective function.
• \(y = \tan(x)\), \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) is an injective function.

Surjective Function

For any \(y\) in the range of the function \(y = f(x)\), there exists \(x\) in the domain such that \(f(x) = y\).

• \(y = x^2\), \(y \in \mathbb{R}\), \(x \in \mathbb{R}\) is not a surjective function.

Composite Function

• Given \(f(x) = x + 1\) and \(g(x) = x^2\), the composite function \(g \circ f(x) = g(f(x)) = (x + 1)^2\).

• The operation \(\circ\) denotes the composition of functions.

• \(f(gh(x)) = (fg)(h(x))\).

• \(f(x) \times f(x) \times f(x) \dots = (f(x))^n\).

• The \(n\)-th derivative is denoted by \(\frac{d^n}{d x^n}f(x) = f^{(n)}(x)\).

• Nested functions can be written as \(f(f(f(\dots x \dots ))) = f^n(x)\).

Inverse Function

• If \(f(x)\) has an inverse, it is denoted as \(f^{-1}(x)\).

• The inverse satisfies \(f(f^{-1}(x)) = f^{-1}(f(x)) = x\).

• A function \(f\) has an inverse if and only if it is a bijective function.

  • Example: \(y = \sin(x)\), \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), \(y \in (-1, 1)\).

• Property: The graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) are symmetric about the line \(y = x\).

Parity of Functions

Even Function

• A function \(g(x): A \rightarrow B\) is even if \[g(x) = g(-x)\] for all \(x \in A\).

Odd Function

• A function \(h(x): A \rightarrow B\) is odd if \[h(x) = -h(-x)\] for all \(x \in A\).

Cubic Function

A cubic function exhibits rotational symmetry of order 2 about the midpoint of the local maxima and local minima.

Periodic Function

Let \(f: A \rightarrow B\). If there exists \(T \in A\) such that \(f(x) = f(x + T)\) for all \(x \in A\), then \(f\) is a periodic function where \(T\) is the period of \(f\).

Properties

• 1. If \(T\) is a period of \(f\), then \(kT\) is also a period of \(f\), where \(k \in \mathbb{Z}\).
• 2. If \(T_1\) and \(T_2\) are periods of \(f\) with \(T_1 \neq T_2\), then $T_1 + \(T_2\) is also a period of \(f\).
  • Example: \(f(x) = f(x + T_1) = f(x + T_1 - T_2)\).

Claim

\(\clubsuit\) Every function \(f(x)\) that is symmetric about \(x = 0\) can be expressed as the sum of an even function and an odd function.

• Define

  \(g(x) = g(-x)\),\(g(x)\) is an even function.

  \(h(x) = -h(-x)\),\(h(x)\) is an odd function.

  \(f(x) = g(x) + h(x)\)

• Proof

  \[ \begin{cases} f(x) = g(x)+h(x) &(1) \\ f(-x) = g(-x)+h(-x) &(2) \end{cases} \] \[ f(-x) = g(x)-h(x)\\ h(x) = f(x)-g(x)=g(x) -f(-x) \] $ g(x) = \ h(x) = $