Learn how to express any function in terms of an even and an odd function[cite: 1]. Step-by-step examples are provided in the form of questions with detailed solutions hidden inside collapsible dropdowns so you can practice independently.
Show that any function \( f(x) \) can be expressed as the sum of an even and an odd function[cite: 1].
We can write \( f(x) \) as: \[ f(x) = \frac{1}{2} f(x) + \frac{1}{2} f(x) + \frac{1}{2} f(-x) - \frac{1}{2} f(-x) = \frac{1}{2} \big(f(x) + f(-x)\big) + \frac{1}{2} \big(f(x) - f(-x)\big) \]
Let \[ g(x) = \frac{1}{2} \big(f(x) + f(-x)\big) \] Check that \( g(x) \) is even: \[ g(-x) = \frac{1}{2} \big(f(-x) + f(x)\big) = g(x) \]
Let \[ h(x) = \frac{1}{2} \big(f(x) - f(-x)\big) \] Check that \( h(x) \) is odd: \[ h(-x) = \frac{1}{2} \big(f(-x) - f(x)\big) = -h(x) \]
Hence, \[ f(x) = g(x) + h(x) \] where \( g(x) \) is even and \( h(x) \) is odd.
Express \[ f(x) = 2x^4 - 5x^3 + 2x^2 + x - 4 \] as the sum of an even and an odd function.
Since \( f(x) \) is a polynomial, we can separate its even and odd parts: \[ f(x) = (2x^4 + 2x^2 - 4) + (-5x^3 + x) \] where \( 2x^4 + 2x^2 - 4 \) is even and \( -5x^3 + x \) is odd.
Express \[ f(x) = \frac{1}{x-1} \] as the sum of an even and an odd function and verify your answer.
From Question 1, any function \( f(x) \) can be expressed as: \[ f(x) = \frac{1}{2} \big(f(x) + f(-x)\big) + \frac{1}{2} \big(f(x) - f(-x)\big) \]
Here, \[ g(x) = \frac{1}{2} \left(\frac{1}{x-1} + \frac{1}{-x-1}\right) = \frac{1}{x^2 - 1} \] \[ h(x) = \frac{1}{2} \left(\frac{1}{x-1} - \frac{1}{-x-1}\right) = \frac{x}{x^2 - 1} \]
Check: \[ g(x) + h(x) = \frac{1}{x^2 - 1} + \frac{x}{x^2 - 1} = \frac{1+x}{x^2-1} = \frac{1+x}{(x-1)(x+1)} = \frac{1}{x-1} = f(x) \]
Questions with Answers[cite: 1], Tutorials and Problems[cite: 1].