Question

a. If \(f(0)\) is defined and \(f\) is an even function, is it necessarily true that \(f(0)=0 ?\) Explain. b. If \(f(0)\) is defined and \(f\) is an odd function, is it necessarily true that \(f(0)=0 ?\) Explain.

Short answer

Answer: It is not necessarily true for even functions, but it is true for odd functions.

Step-by-step solution

Understanding even functions

An even function is a function that satisfies the property \(f(-x) = f(x)\) for all values of \(x\). This means that the function is symmetric about the y-axis, i.e., if we take the mirror image of the graph across the y-axis, the graph remains unchanged.

Check if \(f(0)=0\) for even functions

Since \(f(0)\) is defined, it means that there is a value for the function when \(x=0\). To determine if \(f(0)=0\), we can use the property of even functions: \(f(-x) = f(x)\). Here, let's substitute \(x\) with \(0\). We obtain the equation \(f(-0) = f(0)\). Since \(-0 = 0\), the equation simplifies to \(f(0) = f(0)\). This statement is true, but it doesn't necessarily prove that \(f(0) = 0\). In conclusion, for an even function, it is not necessarily true that \(f(0)=0\). #b. Odd function#

Understanding odd functions

An odd function is a function that satisfies the property \(f(-x) = -f(x)\) for all values of \(x\). This means that the function has rotational symmetry about the origin, i.e., if we rotate the graph \(180^\circ\) about the origin, the graph remains unchanged.

Check if \(f(0)=0\) for odd functions

Since \(f(0)\) is defined, it means that there is a value for the function when \(x=0\). To determine if \(f(0)=0\), we can use the property of odd functions: \(f(-x) = -f(x)\). Here, let's substitute \(x\) with \(0\). We obtain the equation \(f(-0) = -f(0)\). Since \(-0 = 0\), the equation simplifies to \(f(0) = -f(0)\). This statement implies that \(f(0)\) must be equal to zero since it is the only number to be equal to its negative. So, for an odd function, it is necessarily true that \(f(0)=0\).