Problem 20 - 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. $$ \left\\{(x, y) : y=\frac{1}{\sqrt{x+1}}\right\\} $$

Step-by-Step Solution

Answer

a. Domain of set \( A \) is \([-1, \infty)\). b. The function is not onto.

Step 1: Understanding the Function

The given function is defined as \( y = \frac{1}{\sqrt{x+1}} \). To determine the largest possible domain for set \( A \), we need to identify all possible real numbers \( x \) for which this function is defined.

Step 2: Finding the Domain of the Function

The expression \( \sqrt{x+1} \) is defined for all \( x \) such that \( x+1 \geq 0 \). Therefore, \( x \geq -1 \). Hence, the largest subset of the real numbers for \( x \) is \([-1, \infty)\).

Step 3: Evaluating the Function's Codomain

The function outputs \( y = \frac{1}{\sqrt{x+1}} \). \( \sqrt{x+1} \) outputs positive values, so \( y \) will always yield positive real numbers.

Step 4: Determine if the Function is Onto

For the function to be onto, every element of set \( B \) must map from an element in set \( A \). Because the function outputs only positive real numbers and a positive range of \( y \) values, it cannot span all of \( B \) if \( B \) includes any non-positive real number. Hence, the function is not onto when \( A = B \).

Key Concepts

Real Numbers

Real numbers are a critical part of algebra, providing a complete set of numbers including all the integers, rational numbers, and irrational numbers. They fill the entire number line without any gaps. This makes real numbers essential when discussing mathematical concepts like functions.
- Real numbers can be positive, negative, or zero.
- Integers, such as 1, 2, -3, are real numbers.- Fractions and decimals, like \( \frac{1}{2} \) and 0.75, are also real numbers.- Irrational numbers, like \( \pi \) and \( \sqrt{2} \), fall under real numbers.A solid grasp of real numbers is necessary to fully understand domains and codomains in algebraic functions, as they often define and limit these sets.

Domain and Codomain

In functions, the domain and codomain form the backbone of how functions operate. The domain is the set of permissible inputs, while the codomain is the set of potential outputs. These concepts help in understanding the scope and limitations of any function.
- **Domain:** For a function like \( y = \frac{1}{\sqrt{x+1}} \), the domain is all real numbers \( x \) for which the function makes sense. Here, \( x+1 \geq 0 \) translates to \( x \geq -1 \). So, the domain is \([-1, \infty)\).- **Codomain:** These are all the values that the output \( y \) can technically take on. For our function \( y = \frac{1}{\sqrt{x+1}} \), since \( y \) involves a square root, \( y \) can only be positive real numbers.Understanding both domain and codomain helps in predicting and controlling the behavior of functions and identifying their range of effect.

Onto Function

Now, let's talk about whether a function "completely fills" its codomain, otherwise known as being 'onto'. An onto function means every possible output is produced by some input from the domain. It is a crucial part of understanding how functions correlate inputs to outputs.In our example, the function is \( y = \frac{1}{\sqrt{x+1}} \). For it to be onto, every real number in the codomain must be achieved by plugging some real number from the domain into the function. However, because this function's outputs are only positive real numbers, it is not onto if the codomain includes any non-positive number.
Check whether outputs match all possible values in the codomain:- If matches, function is onto.- If not all values are produced, function is not onto.This topic is often crucial in higher math, notably calculus and analysis, as it underpins many theoretical discussions.

Function Definition

In algebra, understanding what a function is can seem quite intricate at first, but it's really about how one set relates to another. A function takes each input from a domain and assigns it to exactly one output in the codomain.Formally, a function from a set \(A\) to a set \(B\) is defined by a rule that assigns every element \(x\) in \(A\) to a unique element \(y\) in \(B\). Let's break that down:- **Unique Mapping:** For each \(x\) in the domain, there is a specific \(y\) from the codomain.- **Well-Defined:** For any input \(x\), the output \(y\) must make sense.In our problem, the function \(y = \frac{1}{\sqrt{x+1}}\) gives clear guidelines:- You can plug in any \(x\) from \([-1, \infty)\).- Each \(x\) gives a unique \(y\), showing the consistent nature of functions.This groundwork is essential as it builds the basis for further explorations in mathematics from simple calculations to complex analyses.