how would you limit the domain to make this function one-to-one and still have the same range?\nf(x) = (x +…

how would you limit the domain to make this function one-to-one and still have the same range?\nf(x) = (x + 3)^2\nx ≥ ?

how would you limit the domain to make this function one-to-one and still have the same range?\nf(x) = (x + 3)^2\nx ≥ ?

Answer

Explanation:

Step1: Analyze the function type

The function ( f(x)=(x + 3)^2 ) is a quadratic function, and its graph is a parabola opening upwards. The vertex of the parabola is at ( x=-3 ) (since for ( y = a(x - h)^2+k ), the vertex is at ( (h,k) ), here ( h=-3,k = 0 )).

Step2: Determine the one - to - one domain

A parabola is symmetric about its vertex. For a parabola opening upwards, to make it one - to - one (pass the horizontal line test), we can take the domain as ( x\geq ) the x - coordinate of the vertex or ( x\leq ) the x - coordinate of the vertex. The range of ( f(x)=(x + 3)^2 ) is ( y\geq0 ). If we want to keep the same range and make the function one - to - one, we take the right - hand side of the vertex (where the function is increasing). The x - coordinate of the vertex is ( - 3 ), so we limit the domain to ( x\geq - 3 ).

Answer:

(-3)