To make the function [tex]\( f(x) = x^2 + 3 \)[/tex] one-to-one and still have the same range, we need to find an appropriate restriction on the domain of [tex]\( x \)[/tex].
1. Understand the Function: The given function is [tex]\( f(x) = x^2 + 3 \)[/tex]. The range of this function is [tex]\([3, \infty)\)[/tex], because [tex]\( x^2 \geq 0 \)[/tex] for all real [tex]\( x \)[/tex], and adding 3 shifts the range to start from 3.
2. Determine Non One-to-One Nature: The function [tex]\( f(x) = x^2 + 3 \)[/tex] is not one-to-one over all real numbers, because for any [tex]\( y \geq 3 \)[/tex], there are generally two possible [tex]\( x \)[/tex] values (one positive and one negative) that satisfy [tex]\( x^2 + 3 = y \)[/tex].
3. Restrict the Domain: To make the function one-to-one while maintaining the same range, we need to restrict the domain to ensure that each [tex]\( y \)[/tex] value in the range [tex]\([3, \infty)\)[/tex] corresponds to only one [tex]\( x \)[/tex]. The simplest way to do this is to limit [tex]\( x \)[/tex] to non-negative values, i.e., [tex]\( x \geq 0 \)[/tex].
4. Verify and Conclude: By restricting the domain to [tex]\( x \geq 0 \)[/tex], each value of [tex]\( y \)[/tex] in [tex]\([3, \infty)\)[/tex] corresponds to exactly one value of [tex]\( x \)[/tex]. Therefore, the range remains the same, and the function becomes one-to-one within this restricted domain.
Thus, to make the function [tex]\( f(x) = x^2 + 3 \)[/tex] one-to-one and maintain the same range, we should restrict the domain as follows:
[tex]\[ x \geq 0 \][/tex]
So, the solution is:
[tex]\[ x \geq 0 \][/tex]
