Even and odd at the origin

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

검증된 단계별 안내

Step 1: Understand the definition of an even function. An even function is one where for every x in the domain, f(x) = f(-x). This means the function is symmetric with respect to the y-axis.

Step 2: Consider the point at the origin, x = 0. For an even function, f(0) should equal f(-0), which simplifies to f(0) = f(0). This is always true, but it doesn't necessarily imply that f(0) = 0.

Step 3: Explore examples of even functions. For instance, the function f(x) = x^2 is even, and f(0) = 0. However, consider another even function, f(x) = x^2 + 1, where f(0) = 1. This shows that f(0) does not have to be 0.

Step 4: Conclude that while f(0) = f(-0) is true for even functions, it does not imply that f(0) must be 0. The value of f(0) depends on the specific function, not just its evenness.

Step 5: Summarize that the statement 'if f(0) is defined and f is an even function, then f(0) = 0' is false. The value of f(0) can be any real number, depending on the function.

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Even Functions

An even function is defined as a function f(x) that satisfies the condition f(-x) = f(x) for all x in its domain. This symmetry about the y-axis implies that the function takes the same value for both positive and negative inputs. A common example is f(x) = x², where f(-x) = (-x)² = x², confirming its evenness.

Function Value at the Origin

The value of a function at the origin, denoted as f(0), is simply the output of the function when the input is zero. For even functions, this value can be any real number, including zero. Thus, knowing that a function is even does not inherently dictate that f(0) must equal zero.

Implications of Evenness at Specific Points

While even functions exhibit symmetry, this property does not impose specific values at particular points unless additional conditions are met. For instance, if f(0) is defined and f is even, it can be any value, not just zero. Therefore, the assertion that f(0) must equal zero is incorrect without further constraints.

Even and odd at the origina. If ƒ(0) is defined and ƒ is an even function, is it necessarily true that ƒ(0) = 0? Explain.