Problem 48 Explain why an even function who... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 1: Problem 48

Explain why an even function whose domain contains a nonzero number cannot be a one-to-one function.

Short Answer

Expert verified

An even function is defined as a function that satisfies the condition f(-x) = f(x) for all x in its domain. For a function to be one-to-one (injective), it must follow that f(x) 鈮� f(y) for all x 鈮� y in its domain. If we let x be a nonzero number in the domain of an even function f, then f(-x) = f(x). This implies that there exist two distinct values of x and -x with equal function values, violating the injective property. Thus, an even function with a nonzero number in its domain cannot be a one-to-one function.

Step by step solution

Defining even function and one-to-one function

An even function is defined by the condition f(-x) = f(x), for all x in its domain. A one-to-one function, also called an injective function, has the property that for any x and y in its domain, if x 鈮� y then f(x) 鈮� f(y). We need to examine whether an even function with a nonzero number in its domain can fulfill this injective property.

Let x be a nonzero number in the domain of the even function

Let x be a nonzero number in the domain of an even function f. By the definition of an even function, we know that f(-x) = f(x).

Show that the injective property does not hold

Now let's consider if f is injective. Since f(-x)=f(x) as derived in step 2, we know that there exist two different inputs, x and -x, such that their function values are the same. In other words, we have that f(x)=f(-x), which means that the injective property, f(x) 鈮� f(y) whenever x 鈮� y, does not hold.

Conclude that an even function with a nonzero in its domain cannot be one-to-one

Since the injective property does not hold for an even function with a nonzero number in its domain, we can conclude that such a function cannot be a one-to-one function. This is because the function values at x and -x will always be equal for an even function, but x and -x are not equal when x is a nonzero number. Thus, an even function that includes a nonzero number in its domain violates the one-to-one property, and cannot be a one-to-one function.

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91影视

Explain why an even function whose domain contains a nonzero number cannot be a one-to-one function.

Short Answer

Step by step solution

Defining even function and one-to-one function

Let x be a nonzero number in the domain of the even function

Show that the injective property does not hold

Conclude that an even function with a nonzero in its domain cannot be one-to-one

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