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Magazine Chapter 1: Problem 4
In Exercises 3 and \(4,\) find the domains and ranges of \(f, g, f / g,\) and \(g / f .\) $$ f(x)=1, \quad g(x)=1+\sqrt{x} $$
Short Answer
Expert verified
Domains: \( f: (-\infty, \infty), g: [0, \infty), f/g: [0, \infty), g/f: [0, \infty) \). Ranges: \( f: \{1\}, g: [1,\infty), f/g: (0,1], g/f: [1,\infty) \).
Step by step solution
01
Finding the Domain of f(x)
The function \( f(x) = 1 \) is a constant function. Constant functions have domains that include all real numbers because there is no restriction on the input values. Thus, the domain of \( f(x) \) is \( (-\infty, \infty) \).
02
Finding the Range of f(x)
Since \( f(x) = 1 \) is a constant function, its output is always \( 1 \) regardless of the input. Therefore, the range of \( f(x) \) is \( \{ 1 \} \).
03
Finding the Domain of g(x)
The function \( g(x) = 1 + \sqrt{x} \) is defined only for non-negative values of \( x \) because the square root function requires non-negative input in real numbers. Therefore, the domain of \( g(x) \) is \( [0, \infty) \).
04
Finding the Range of g(x)
For \( g(x) = 1 + \sqrt{x} \), as \( x \) increases from 0 to \( \infty \), the square root function also increases from 0 to \( \infty \). Thus, the range of \( g(x) \) starts from \( 1 + 0 = 1 \) to \( \infty \). Hence, the range is \( [1, \infty) \).
05
Finding the Domain of f/g
The function \( \frac{f(x)}{g(x)} = \frac{1}{1+\sqrt{x}} \) is defined as long as \( g(x) = 1 + \sqrt{x} eq 0 \). Since \( g(x) \) is always at least 1, the domain for \( \frac{f}{g} \) is the same as the domain of \( g(x) \), which is \( [0, \infty) \).
06
Finding the Range of f/g
For \( \frac{f(x)}{g(x)} = \frac{1}{1+\sqrt{x}} \), as \( \sqrt{x} \) goes from 0 to \infty, \( 1 + \sqrt{x} \) goes from 1 to \infty, making \( \frac{1}{1+\sqrt{x}} \) reduce from 1 to 0. Thus, the range is \((0, 1]\).
07
Finding the Domain of g/f
The function \( \frac{g(x)}{f(x)} = \frac{1+\sqrt{x}}{1} = 1 + \sqrt{x} \) shares its domain with \( g(x) \). Therefore, its domain is \( [0, \infty) \).
08
Finding the Range of g/f
Since \( \frac{g(x)}{f(x)} = 1 + \sqrt{x} \) and it seems unaltered by the division by \( f(x) \), the range of \( g/f \) remains the range of \( g(x) \), which is \( [1, \infty) \).
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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 type of mathematical function that always returns the same value no matter what the input is. This means it doesn't matter which number you substitute for the variable; the output remains unchanged. It is represented as \( f(x) = c \), where \( c \) is a constant value.
When analyzing the domain and range of a constant function:
When analyzing the domain and range of a constant function:
- Domain: The domain is all real numbers \((-\infty, \infty)\) because there are no restrictions on what \( x \) can be.
- Range: The range is always \( \{ c \} \), as the function only outputs one value, \( c \).
Square Root Function
A square root function involves the square root operation, which can be represented as \( g(x) = a + \sqrt{x} \), where \( a \) is a constant. The function discussed here, \( g(x) = 1 + \sqrt{x} \), showcases how square root operations impose constraints on function domains and ranges.
- Domain: The domain is limited to non-negative values of \( x \) (i.e., \( [0, \infty) \)) because you cannot take the square root of a negative number in the realm of real numbers.
- Range: This function increases gradually and has no upper limit, starting from \( 1 \) and moving towards \( \infty \). Hence the range is \( [1, \infty) \).
Domain and Range Analysis
Understanding a function's domain and range is essential in expressing the valid inputs and possible outputs. Domains specify all the input values \( x \) for which the function \( f(x) \) is defined. Ranges provide the corresponding output values a function can produce.
When analyzing domain and range:
When analyzing domain and range:
- Consider any restrictions such as divides by zero or square roots of negative numbers.
- Observe how the function behaves as \( x \) approaches different boundaries.
- Look closely at transformations, such as shifts or stretches, affecting range but not necessarily domain.
Function Operations
Function operations involve combining two or more functions to produce a new one, through addition, subtraction, multiplication or division.
With division, like \( \frac{f(x)}{g(x)} \), it's essential to manage domains carefully to avoid undefined operations (e.g., division by zero).
Here, both \( \frac{f}{g} \) and \( \frac{g}{f} \), derive respective domains and ranges by combining the rules of the original functions:
With division, like \( \frac{f(x)}{g(x)} \), it's essential to manage domains carefully to avoid undefined operations (e.g., division by zero).
Here, both \( \frac{f}{g} \) and \( \frac{g}{f} \), derive respective domains and ranges by combining the rules of the original functions:
- For \( \frac{f}{g} \): The domain is the same as \( g(x) \) because \( g \) should not be zero, resulting in a domain of \( [0, \infty) \). The range reduces as \( \sqrt{x} \) increases, giving \( (0, 1] \).
- For \( \frac{g}{f} \): It mirrors the domain and range of \( g(x) \) due to the division by a constant, resulting in a domain of \( [0, \infty) \) and range \([1, \infty) \).
