Problem 25
Question
For each function, determine the largest possible domain. (a) \(f(x)=\frac{1}{x^{2}+2 x+1}\) (b) \(g(x)=\sqrt{x^{2}+2 x+1}\) Factoring will help clarify the solution.
Step-by-Step Solution
Verified Answer
The domain of \(f(x)\) is all real numbers except -1 and the domain of \(g(x)\) is all real numbers.
1Step 1: Find the Domain for \(f(x)\)
To find the domain of \(f(x)\), it is crucial to set the denominator equal to zero and solve, as the function is undefined for those values of x. Therefore, solving for \(x^2 + 2x + 1 = 0\) will yield the values to be excluded from the domain. Here, \(x = -1\), which means the domain of \(f(x)\) is all real numbers except -1.
2Step 2: Find the Domain for \(g(x)\)
For \(g(x)\), the expression inside the square root must be greater than or equal to zero. Hence, \(x^2 + 2x + 1 \geq 0\). This inequality holds for all real numbers, so the domain of \(g(x)\) is all real numbers.
Key Concepts
Determining Function DomainContinuous FunctionFunctions and Their Domains
Determining Function Domain
Understanding the domain of a function is essential for correctly solving and graphing equations. Essentially, the domain refers to all the possible input values that a function can accept without causing any mathematical anomalies such as division by zero or the square root of a negative number.
In the exercise given, for instance, function f(x) has a denominator of x^2 + 2x + 1. When determining the domain for f(x), it is essential to ensure that this denominator never equals zero, since division by zero is undefined. By setting x^2 + 2x + 1 = 0 and solving for x, we find that x = -1 should be excluded from the domain. Therefore, the largest possible domain of f(x) is all real numbers except x = -1.
While determining domains, it's always important to factor accordingly, as suggested in the exercise -- it can help to clarify solutions, by revealing potential restrictions in a more straightforward manner.
In the exercise given, for instance, function f(x) has a denominator of x^2 + 2x + 1. When determining the domain for f(x), it is essential to ensure that this denominator never equals zero, since division by zero is undefined. By setting x^2 + 2x + 1 = 0 and solving for x, we find that x = -1 should be excluded from the domain. Therefore, the largest possible domain of f(x) is all real numbers except x = -1.
While determining domains, it's always important to factor accordingly, as suggested in the exercise -- it can help to clarify solutions, by revealing potential restrictions in a more straightforward manner.
Continuous Function
A continuous function, in mathematical terms, is a function that does not have any interruptions, jumps, or breaks in its graph. Intuitively, if you can draw the graph of a function without lifting your pen off the paper, the function is continuous. For a function to be continuous, every value within the domain must correspond to a value in the range.
In the context of function g(x) from our exercise, we look at the definition involving square roots. The function under the square root, x^2 + 2x + 1, must be non-negative for all real numbers in the domain because the square root of a negative number is not a real number. Fortunately, the expression can be factored into (x+1)^2, which is always greater than or equal to zero for any real number x. This means the function g(x) is continuous over the whole set of real numbers, making its domain all real numbers.
In the context of function g(x) from our exercise, we look at the definition involving square roots. The function under the square root, x^2 + 2x + 1, must be non-negative for all real numbers in the domain because the square root of a negative number is not a real number. Fortunately, the expression can be factored into (x+1)^2, which is always greater than or equal to zero for any real number x. This means the function g(x) is continuous over the whole set of real numbers, making its domain all real numbers.
Functions and Their Domains
Functions are one of the fundamental concepts in mathematics used for depicting the relationship between sets of information. Each function has a domain and a range. While the domain encompasses all the input values (x-values) for the function, the range consists of all the output values (y-values) that result from those inputs.
In the provided exercise, both functions, f(x) and g(x), include the same quadratic expression, but their domains are different due to the operations involved. It’s important to analyze a function's structure to understand its domain correctly: look for divisions, square roots, or logarithms, as these operations impose specific conditions on x-values that are allowed.
In the provided exercise, both functions, f(x) and g(x), include the same quadratic expression, but their domains are different due to the operations involved. It’s important to analyze a function's structure to understand its domain correctly: look for divisions, square roots, or logarithms, as these operations impose specific conditions on x-values that are allowed.
- For division, exclude values that make the denominator zero.
- For square roots, include only values that make the expression inside the root greater than or equal to zero.
- For logarithms, only positive values are permissible.
Other exercises in this chapter
Problem 23
For each function, determine the largest possible domain. (a) \(f(x)=\frac{1}{x^{2}-1}\) (b) \(g(x)=\sqrt{x^{2}-1}\) For part (b), factor the quadratic. The pro
View solution Problem 24
For each function, determine the largest possible domain. (a) \(f(x)=\frac{3}{x^{2}+3 x-4}\) (b) \(g(x)=\sqrt{x^{2}+3 x-4}\) Factoring will help clarify the sol
View solution Problem 26
Express the area of a circle as a function of: (a) its diameter, \(d\). (b) its circumference, \(c .\)
View solution Problem 26
For each function, determine the largest possible domain. (a) \(f(x)=\frac{1}{x+2}-\frac{1}{x-1}\) (b) \(g(x)=\sqrt{x+2}-\sqrt{x-1}\)
View solution