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Magazine Chapter 1: Problem 3
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
Domains: \((-\infty, \infty)\) for all functions; Ranges: \(\{2\}\), \([1,\infty)\), \((0,2]\), \([\frac{1}{2},\infty)\).
Step by step solution
01
Find the Domain of f(x)
The function \(f(x) = 2\) is a constant function, meaning it assigns the value 2 to any input \(x\). There are no restrictions or conditions on the variable \(x\). Therefore, the domain of \(f(x)\) is all real numbers, which can be expressed as \( (-\infty, \infty) \).
02
Find the Domain of g(x)
The function \(g(x) = x^2 + 1\) is a polynomial function. Polynomial functions are defined for all real numbers. Thus, the domain of \(g(x)\) is also all real numbers, expressed as \( (-\infty, \infty) \).
03
Find the Range of f(x)
Since \(f(x)\) is a constant function that always equals 2, the range of \(f(x)\) is simply the set containing the number 2, expressed as \( \{ 2 \} \).
04
Find the Range of g(x)
The function \(g(x) = x^2 + 1\) will always result in values that are 1 or greater because \(x^2\) is non-negative (\(x^2 \geq 0\)) and this value is increased by 1. Therefore, the range of \(g(x)\) is \([1, \infty)\).
05
Find the Domain of \(\frac{f}{g}\)
To find the domain of \( \frac{f}{g}(x) = \frac{2}{x^2 + 1} \), we need to ensure that the denominator is not zero. However, since \(x^2 + 1\) is always greater than or equal to 1 for all real \(x\), the denominator never becomes zero. Therefore, the domain is all real numbers \((-\infty, \infty)\).
06
Find the Domain of \(\frac{g}{f}\)
The function \( \frac{g}{f}(x) = \frac{x^2 + 1}{2} \) requires that the denominator is not zero. Since 2 is a constant and never zero, the domain of \( \frac{g}{f} \) is all real numbers \((-\infty, \infty)\).
07
Find the Range of \(\frac{f}{g}\)
The range of \( \frac{f}{g}(x) = \frac{2}{x^2 + 1} \) is determined by the minimum and maximum values of the expression \(x^2 + 1\). The smallest value is 1 (when \(x=0\)), resulting in \( \frac{2}{1} = 2 \). The expression decreases toward 0 as \(x\) becomes very large, allowing \( \frac{2}{x^2 + 1} \) to get infinitesimally close to 0 but never reach it. Thus, the range is \((0, 2]\).
08
Find the Range of \(\frac{g}{f}\)
The function \( \frac{g}{f}(x) = \frac{x^2 + 1}{2} \) essentially scales the range of \(g(x)\) by 1/2. Originally, \(g(x)\) ranges from 1 to \( \infty\), so \( \frac{g}{f}(x) \) ranges from \( \frac{1}{2} \) to \( \infty \). The range is therefore \([\frac{1}{2}, \infty)\).
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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 a function that always returns the same value, no matter what the input is. In mathematical terms, if we have a function \( f(x) = c \), where \( c \) is a constant, the output is always \( c \) for any input \( x \).
This means constant functions have very simple domains and ranges:
This means constant functions have very simple domains and ranges:
- Domain: The domain of a constant function is all real numbers \((-\infty, \infty)\) because we can put any real number into the function.
- Range: The range of a constant function is just the constant value \( \{c\} \). For example, if \( f(x) = 2 \), the range is \( \{2\} \).
Polynomial Function
Polynomial functions are a type of mathematical expression that involve variables raised to whole number powers and coefficients. The general form of a polynomial function is:\[ g(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer.
Key characteristics of polynomial functions include:
Key characteristics of polynomial functions include:
- Domain: Polynomial functions are defined for all real numbers, \((-\infty, \infty)\), meaning there are no restrictions on the inputs we can choose for them.
- Range: The range of a polynomial function depends on the degree and leading coefficient, and it can vary widely. For example, the function \( g(x) = x^2+1 \) has a range of \([1, \infty)\).
Rational Function
Rational functions are created by dividing two polynomial functions. They take the form \( \frac{f(x)}{g(x)} \) where both \( f(x) \) and \( g(x) \) are polynomials.
Some important aspects to be aware of include:
Some important aspects to be aware of include:
- Domain: The domain of a rational function excludes values that make the denominator zero. For a function like \( \frac{2}{x^2 + 1} \), since \( x^2 + 1 \) never equals zero, the domain includes all real numbers \( (-\infty, \infty) \).
- Range: The range can be more complex to determine. For \( \frac{2}{x^2 + 1} \), the range is \( (0, 2] \), since the expression approaches 0 as \( x \) becomes very large and reaches 2 when \( x = 0 \).
Real Numbers
Real numbers include all the numbers on the standard number line. This broad category of numbers encompasses:
- Integers: Both positive and negative whole numbers, including zero (e.g., -3, 0, 5).
- Rational Numbers: Numbers that can be expressed as a fraction of integers (e.g., \( \frac{1}{2} \), \( -\frac{3}{4} \)).
- Irrational Numbers: Numbers that cannot be expressed as simple fractions (e.g., \( \sqrt{2} \), \( \pi \)).
- Concept of Continuity: The real numbers have no gaps, meaning they form a continuous line from negative infinity to positive infinity.
- Uses in Functions: In mathematical functions, real numbers often represent possible inputs and outputs, allowing functions such as polynomials to be defined across the entirety of their domain.
