Question

consider the quadratic function
y = x² - 4
which statement is true?
the function is not one - to - one because each value in the range of the function maps to a unique value in the domain of the function
the function is not one - to - one because some values in the range of the function map to more than one value in the domain of the function
the function is one - to - one because each value in the range of the function maps to a unique value in the domain of the function and vice versa
the function is one - to - one because some values in the range of the function map to more than one value in the domain of the function

Explanation:

Step1: Recall one - to - one function definition

A function \(y = f(x)\) is one - to - one (injective) if for every \(y\) in the range of \(f\), there is exactly one \(x\) in the domain such that \(y = f(x)\). In other words, if \(f(a)=f(b)\), then \(a = b\) for all \(a,b\) in the domain.

For the quadratic function \(y=x^{2}-4\), let's assume \(f(a)=f(b)\). Then \(a^{2}-4=b^{2}-4\). Subtracting \(- 4\) from both sides gives \(a^{2}=b^{2}\), which implies \(a = b\) or \(a=-b\). For example, if \(x = 2\) and \(x=-2\), \(f(2)=2^{2}-4 = 0\) and \(f(-2)=(-2)^{2}-4 = 0\). So, different \(x\) values (2 and - 2) map to the same \(y\) value (0). This means that some values in the range of the function map to more than one value in the domain. So the function is not one - to - one.

Step2: Analyze each option

• Option 1: Says the function is not one - to - one because each value in the range maps to a unique value in the domain. This is incorrect. We saw that a single \(y\) value can map to multiple \(x\) values.
• Option 2: Says the function is not one - to - one because some values in the range map to more than one value in the domain. This matches our analysis.
• Option 3: Says the function is one - to - one because each range value maps to a unique domain value. But we know from the example above that this is false.
• Option 4: Says the function is one - to - one because some range values map to more than one domain values. The definition of one - to - one is that each range value maps to exactly one domain value. So this is incorrect.

Answer:

The option "The function is not one - to - one because some values in the range of the function map to more than one value in the domain of the function" (the second option among the given choices)