Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![### Functions and Domains
In this exercise, we will be working with mathematical functions and their respective domains. We will also find the composition of functions and determine their domains, all reported in interval notation.
#### Problem 16
**Given:**
\[ f(x) = -x^3 + 3x - 4 \]
\[ g(x) = 2 - x \]
**Task:**
Find the composition \( (f \circ g)(x) \) and its domain. Report the domain in interval notation.
#### Problem 17
**Given:**
\[ f(x) = \frac{3x+2}{x^2 - 16} \]
\[ g(x) = \frac{5x-4}{x^2 - 16} \]
**Task:**
Determine the domains of the given functions \( f(x) \) and \( g(x) \) and report them in interval notation in the space provided.
### Explanation of Concepts:
**Composition of Functions:**
The composition \( (f \circ g)(x) \) means that \( g(x) \) is substituted into \( f(x) \). Mathematically, this is written as:
\[ (f \circ g)(x) = f(g(x)) \]
**Domains in Interval Notation:**
To find the domain of a function, we determine the set of all possible input values (x-values) that will produce a valid output (y-value). The domain can then be expressed in interval notation, which uses parentheses and brackets to describe a set of numbers. For example, the domain of a function might be all real numbers except those that make the denominator zero in a rational function.
#### Domains of Rational Functions:
For rational functions, the domain excludes values that make the denominator zero. Hence, for \( f(x) = \frac{3x+2}{x^2 - 16} \), we must solve \( x^2 - 16 = 0 \) to find the excluded values.
### Steps to Solve:
1. **Find \( (f \circ g)(x) \):**
Substitute \( g(x) \) into \( f(x) \) and simplify the expression.
2. **Determine the domain of \( (f \circ g)(x) \):**
Identify any restrictions from both \( f(x) \) and \( g(x) \) that affect the domain of the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77a13500-3283-4aa2-b558-e9fbf21d692d%2F11d3d4d0-1b47-4f9a-8c90-d6883fd7c6cf%2Fpz2njw_processed.jpeg&w=3840&q=75)
Step by step
Solved in 3 steps with 2 images
