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Magazine Chapter 1: Problem 3
In Exercises 3 and \(4,\) find the domains and ranges of \(f, g, f / g,\) and \(g / f\) $$ f(x)=2, \quad g(x)=x^{2}+1 $$
Short Answer
Expert verified
Domain of \( f(x) \) and \( g(x) \): \( \mathbb{R} \); Range of \( f(x) \): \{2\}; Range of \( g(x) \): [1, \infty); Domain of \( f/g \) and \( g/f \): \( \mathbb{R} \); Range of \( f/g \): (0, 2]; Range of \( g/f \): [\frac{1}{2}, \infty).
Step by step solution
01
Identify the Domain of f(x)
The function \( f(x) = 2 \) is a constant function. Since there are no restrictions on \( x \) (such as denominators or square roots that could be undefined), the domain of \( f(x) \) is all real numbers, \( \mathbb{R} \).
02
Identify the Range of f(x)
For the function \( f(x) = 2 \), regardless of the input, the output is always \( 2 \). Therefore, the range of \( f(x) \) is \( \{2\} \).
03
Identify the Domain of g(x)
The function \( g(x) = x^2 + 1 \) is a polynomial function. Polynomial functions have a domain of all real numbers since they are defined for every \( x \) in \( \mathbb{R} \).
04
Identify the Range of g(x)
The function \( g(x) = x^2 + 1 \) is the sum of a square term and a constant. The square term \( x^2 \) is always non-negative, with a minimum value of \( 0 \). Thus, \( x^2 + 1 \) has a minimum value of \( 1 \). Therefore, the range of \( g(x) \) is \([1, \infty)\).
05
Identify the Domain of f/g
The function \( \frac{f(x)}{g(x)} = \frac{2}{x^2 + 1} \) is a fraction. The denominator \( x^2 + 1 \) equals zero when \( x = 0 \), which is not the case since \( x^2 + 1 \geq 1 \). Therefore, the domain is all real numbers, \( \mathbb{R} \).
06
Identify the Range of f/g
The range is determined by considering the resulting values from \( \frac{2}{x^2 + 1} \). Since \( x^2 + 1 \) varies from 1 to infinity, the fraction varies from \( \frac{2}{1} = 2 \) to \( 0 \), never actually reaching \( 0 \). So, the range is \((0, 2] \).
07
Identify the Domain of g/f
The function \( \frac{g(x)}{f(x)} = \frac{x^2 + 1}{2} \) is also a fraction, where the denominator is \( 2 \), a constant, and never zero. Therefore, the domain is all real numbers, \( \mathbb{R} \).
08
Identify the Range of g/f
For the range of \( \frac{x^2 + 1}{2} \), consider that \( x^2 + 1 \) has a range of \([1, \infty)\). Therefore, \( \frac{x^2 + 1}{2} \) ranges from \( \frac{1}{2} \) to infinity. Thus, the range is \([\frac{1}{2}, \infty)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Constant Function
A constant function is one of the simplest types of functions you can encounter. It is defined as a function where the output value remains the same regardless of the input value. In mathematical terms, if a function is given by \( f(x) = c \), where \( c \) is a constant, then it is a constant function.
For such functions, understanding both the domain and range is straightforward:
For such functions, understanding both the domain and range is straightforward:
- Domain: The domain of a constant function is all real numbers, \( \mathbb{R} \). This is because there are no operations that could restrict the values of \( x \). The function is defined for every real number.
- Range: The range of a constant function is a single value \( \{ c \} \). Since the output does not change regardless of the input, the entire range is just the constant value \( c \).
Polynomial Function
Polynomial functions are quite prevalent in mathematics and daily applications, representing a broader class of functions. These functions are characterized by the algebraic sum of terms of the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n, a_{n-1}, \ldots \), and \( a_0 \) are constants, and \( n \) is a non-negative integer.
In the context of domain and range:
In the context of domain and range:
- Domain: The domain of a polynomial function is always all real numbers, \( \mathbb{R} \). This is because polynomial expressions are continuous and defined for every real input.
- Range: The range of a polynomial function can vary based on its degree. For example, our polynomial \( g(x) = x^2 + 1 \) has a range \( [1, \infty) \). This is because \( x^2 \) is always non-negative, and adding 1 shifts the minimum value to 1.
Fractional Function
Fractional functions involve a scenario where one function is divided by another, forming a fraction. They are expressed in the form \( \frac{f(x)}{g(x)} \), where \( f(x) \) and \( g(x) \) are other functions.
Understanding the domain involves ensuring the denominator never equals zero:
Understanding the domain involves ensuring the denominator never equals zero:
- Domain: The domain of a fractional function is all real numbers for which the denominator \( g(x) \) is not zero. In our example \( \frac{2}{x^2 + 1} \), the denominator is always at least 1. Hence, the domain is all real numbers, \( \mathbb{R} \).
- Range: The range can be determined by evaluating the values the fraction can produce. For \( \frac{2}{x^2 + 1} \), the output varies between \( 2 \) (when \( x=0 \)) and approaches \( 0 \) as \( x \) increases or decreases indefinitely. Thus, the range is \((0, 2] \).
