Problem 19 - Algebra 2 and Trigonometry

Question

In \(12-23,\) each set is a function from set \(A\) to set \(B .\) a. What is the largest subset of the real numbers that can be set \(A\) , the domain of the given function? b. If set \(A=\operatorname{set} B,\) is the function onto? Justify your answer. $$ \\{(x, y) : y=\sqrt{3-x}\\} $$

Step-by-Step Solution

Answer

a) Domain \( A = [ -\infty, 3 ] \). b) Yes, the function is onto in this case.

Step 1: Identify the domain of the function

The function given is \( y = \sqrt{3 - x} \). For the square root to be defined over the real numbers, the expression inside the square root, \(3 - x\), must be non-negative. This leads us to the inequality \(3 - x \geq 0\). Solving for \(x\), we get \(x \leq 3\). Thus, the domain is the largest subset of real numbers \([ -\infty, 3 ]\) that satisfy this condition.

Step 2: Determine the range (codomain) of the function

The range of the function \( y = \sqrt{3 - x} \) is determined by the possible values of \( y \). Since \( y \) is a square root, \( y \geq 0 \) for all \( x \) in the domain. Maximum value of \( y \) occurs when \( x = 3 \), giving \( y = 0 \). As \( x \) approaches \(-\infty\), \( y \) approaches 3. Therefore, the range is \([0, 3]\).

Step 3: Analyze if the function is onto when \( A = B \)

A function is onto if every element in the codomain is mapped to by some element in the domain. Here, if \( A = B \), and \( A \) is the domain \([ -\infty, 3 ]\), and \( B \) is the range \([0, 3]\), the function is indeed onto because the range \([0, 3]\) exactly matches the codomain, meaning every \( y \) value in the codomain \([0,3]\) is achievable.

Key Concepts

Domain of a Function

The domain of a function refers to all possible input values (typically "x" values) that allow the function to work without any mathematical hiccups. In simple terms, it's the set of all possible numbers that you can put into the function and still get a sensible output. For a function like \(y = \sqrt{3 - x}\), a square root must have a non-negative value inside it to be defined for real numbers. This requires that \(3 - x \geq 0\). Solving this inequality, we get \(x \leq 3\). Therefore, the largest subset of the real numbers that can be set \(A\), the domain of this function, is \([ -\infty, 3 ]\). In this interval, every \(x\) from negative infinity up to and including 3 can be plugged into the function without running into any 'math errors.' With functions, especially those with square roots, always pay close attention to where the function may become undefined and solve for those points.

Range of a Function

While the domain deals with possible input values, the range of a function is all about the possible output values, or the "y" values, you can get from the function. For the function \(y = \sqrt{3 - x}\), we need to look at what values \(y\) can take on.Since the square root function only gives non-negative outputs, \(y\) will always be greater than or equal to 0. If we analyze the maximum and minimum values, when \(x=3\), \(y=0\). As \(x\) decreases beyond 3 towards negative infinity, \(y\) increases towards 3 but will never exceed it. Hence, the range of this function is \([0, 3]\). Understanding both the domain and range provides a complete picture of how a function behaves within its defined parameters, giving insight into the smallest and largest values the function can achieve.

Onto Function

An onto function, also known as a surjective function, is a type of function where every element in the codomain is mapped to by some element from the domain. In the case where we have a function \(y = \sqrt{3 - x}\), and we are given that both the domain and the codomain are \([0, 3]\), we need to check if each element of the codomain has a corresponding element in the domain that maps to it.In our setup, the codomain (set \(B\), or the range \([0, 3]\)) is exactly the same as the range of the function. This means that every possible output \(y\) value from 0 to 3 is accounted for and is hit by some input \(x\) within the domain. Thus, every element of the codomain has at least one corresponding element in the domain mapping to it. Because the function's range fully covers the codomain, the function is indeed onto or surjective. An onto function ensures that there are no unused spots or 'gaps' in the codomain, effectively distributing the domain's outputs across the entire codomain.