Problem 59 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the domain of \(g(x)=3\left|x^{2}-1\right|-5\)

Step-by-Step Solution

Answer

The domain is all real numbers, (- , ).

Step 1: Understand the function

The function given is: g(x) = 3|x^2 - 1| - 5 This function involves an absolute value expression.

Step 2: Identify restrictions

Determine if there are any restrictions on the domain. The expression inside the absolute value is a polynomial: x^2 - 1 Polynomials are defined for all real numbers.

Step 3: Specify the domain

Since the expression inside the absolute value is defined for all real numbers and there are no other operations that could restrict the domain (such as division by zero or even roots of negative numbers), the domain is all real numbers.

Key Concepts

Absolute Value Function

An absolute value function takes any input and returns its non-negative value. In mathematical terms, the absolute value of a number \(x\) is written as \(|x|\). It is defined as:

• If \(x \geq 0\), then \(|x| = x\)
• If \(x < 0\), then \(|x| = -x\)

So, when you see an absolute value in a function, like \(|x^2 - 1|\), it means that whether \(x^2 - 1\) is positive or negative, the function will always take the positive value of \(x^2 - 1\). This characteristic is key in simplifying and analyzing these functions.
The function in our example is \(g(x) = 3|x^2 - 1| - 5\). The absolute value part ensures that \(|x^2 - 1|\) is always non-negative, so we never need to worry about it being undefined.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, \(x^2 - 1\) in our function \(g(x) = 3|x^2 - 1| - 5\) is a simple polynomial.
Generally, polynomials can be written as
\[a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\]
where \(a_n, a_{n-1},...,a_0\) are constants and \(n\) is a non-negative integer.
Polynomials are important in this exercise because they are defined for all real numbers. This means there are no values of \(x\) that would make the polynomial undefined. As a result, in the function \(g(x)\), we do not need to exclude any values of \(x\) based on the polynomial part.

Real Numbers

Real numbers include all the numbers that can be found on the number line. This includes both rational numbers (like fractions 3/4 or -2) and irrational numbers (like square root of 2 or \(\pi\)).
When we talk about the domain of a function, we are usually interested in identifying the set of all possible input values (\(x\)-values) for which the function is defined.
For the function \(g(x) = 3|x^2 - 1| - 5\), because the internal polynomial \(x^2 - 1\) is defined for all real numbers, and the absolute value operation and subtraction of 5 do not impose any restrictions, the domain of \(g(x)\) is all real numbers.
In mathematical terms, this is written as: \text{Domain of } g(x) = (-\infty, +\infty)\.