Function as Mapping Math Example 4

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Example 4

medium
Let f:R→Rf: \mathbb{R} \to \mathbb{R}, f(x)=x2f(x) = x^2. Find f−1({4})f^{-1}(\{4\}) (the pre-image of 44) and explain why ff does not have an inverse function on all of R\mathbb{R}.

Solution

  1. 1
    Find all xx such that f(x)=4f(x) = 4: x2=4⇒x=±2x^2 = 4 \Rightarrow x = \pm 2. So f−1({4})={−2,2}f^{-1}(\{4\}) = \{-2, 2\}.
  2. 2
    Since two distinct inputs map to the same output, ff is not one-to-one. An inverse function would require assigning a unique pre-image to 44, but there are two choices. Thus ff has no inverse on all of R\mathbb{R}.

Answer

f−1({4})={−2,2}f^{-1}(\{4\}) = \{-2, 2\}; ff has no inverse on R\mathbb{R} (not one-to-one)
An inverse function exists only when the original function is one-to-one (injective). Because f(x)=x2f(x)=x^2 maps both 22 and −2-2 to 44, we cannot unambiguously reverse the mapping.

About Function as Mapping

Viewing a function as a mapping means thinking of it as an explicit association from each element of the domain to exactly one element of the codomain.

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More Function as Mapping Examples

Example 1 easy

Let [formula] be defined by [formula], [formula], [formula]. Determine whether [formula] is a valid

Example 2 medium

Explain why the relation [formula] is NOT a function from [formula] to [formula].

Example 3 easy

Which of the following sets of ordered pairs defines a function from [formula] to [formula]? (A) [fo

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