Problem 16 - 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. $$ \\{(x, y) : y=\sqrt{2 x}\\} $$

Step-by-Step Solution

Answer

a. Domain: \([0, \infty)\). b. The function is onto since its range matches set \(B\).

Step 1: Identify the function

The function given is defined by the relation \( y = \sqrt{2x} \). This tells us that the function outputs the square root of \(2x\) for every input \(x\).

Step 2: Determine the domain

The domain of a function is the set of all possible inputs. Since the function involves a square root, we need the expression under the square root, \(2x\), to be non-negative (i.e., \(2x \geq 0\)). Simplifying gives \(x \geq 0\). Thus, the largest subset of the real numbers for the domain of the function is \([0, \infty)\).

Step 3: Check if the function is onto given \(A = B\)

For a function to be onto, every element in set \(B\) (the codomain) must be achievable by some element in set \(A\). According to the problem, \(A = B\). From Step 2, we know \(A = [0, \infty)\). Therefore, \(B\) must also be \([0, \infty)\). The function outputs \( y = \sqrt{2x} \), where \(x\) starts from 0 to infinity. The range of \(y\) is also \([0, \infty)\), meaning every element in \(B\) is achieved, so the function is onto.

Key Concepts

Domain of a Function

In mathematics, the domain of a function refers to the complete set of possible input values ("x" values) that the function can accept without resulting in undefined or nonsensical outcomes.
For example, imagine a vending machine programmed to accept only coins. If you try to use a bill, the machine won't function correctly because it's outside the domain of acceptable currency.

For the function given by the equation: - \( y = \sqrt{2x} \) it’s clear there's a square root involved. The square root function is only defined for non-negative numbers, as you can't find the square root of a negative number in the set of real numbers without stepping into more advanced math (like complex numbers).
Thus, to keep the expression \( \sqrt{2x} \) valid, \( 2x \) must be greater than or equal to 0. If we solve for \( x \), we get \( x \geq 0 \).

This means the largest subset of the real numbers that serves as a valid domain is \([0, \infty)\). Anything less than 0 would cause the function to try and compute a square root of a negative number—an operation that is not allowed in this context.

Range of a Function

The range of a function is the set of all possible output values ("y" values) that the function can produce based on its domain.
Using the vending machine analogy, if the machine gives out only certain types of snacks, then the types of snacks represent the range of outputs you can get.

For the function \( y = \sqrt{2x} \): - Since \( x \) belongs to the domain \([0, \infty)\), each calculation of \( y \) results in a non-negative number starting from 0 because square roots of positive numbers are always positive, and square roots of zero is zero itself.

This leads us to the conclusion that the range of the function is also \([0, \infty)\). In simpler terms, the output \( y \) is capable of achieving any value from 0 to infinity, but it cannot be negative.

Onto Function

An onto function, also known as a surjective function, is one where every element of the function's codomain (set \( B \)) is mapped to by at least one element of its domain (set \( A \)). Imagine a coat rack where every hook (codomain) has at least one coat (domain) hanging on it. There are no empty hooks.

In the exercise, since set \( A \) is equal to set \( B \), and both have the interval \([0, \infty)\), we want to determine if our function \( y = \sqrt{2x} \) covers every possible value of the codomain.

Since \( y \) values result directly from \( x \) values that are non-negative, starting from zero and extending upwards, every possible value in the codomain from 0 upwards can be achieved.
Hence, since there's a way to obtain every element of \([0, \infty)\) as an output from an input \( x \) within the same range, this function is indeed onto. Ensuring all hooks have a coat!