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In Exercises 11–26, determine whether each equation defines y as a function of x. x + y = 16
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Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 3, Problema 11a
Find the domain of each function. f(x) = 1/(x2+1) - 1/(x2-1)
Guida verificata passo dopo passo1
Step 1: Recall that the domain of a function consists of all real numbers for which the function is defined. For rational functions, the function is undefined when the denominator equals zero. Identify the denominators in the given function: \( f(x) = \frac{1}{x^2+1} - \frac{1}{x^2-1} \).
Step 2: Analyze the first term \( \frac{1}{x^2+1} \). The denominator \( x^2+1 \) is never zero for any real value of \( x \), because \( x^2+1 \geq 1 \) for all real \( x \). Therefore, this term does not restrict the domain.
Step 3: Analyze the second term \( \frac{1}{x^2-1} \). The denominator \( x^2-1 \) equals zero when \( x^2 = 1 \). Solve \( x^2 = 1 \) to find \( x = \pm 1 \). These values make the denominator zero, so \( x = 1 \) and \( x = -1 \) must be excluded from the domain.
Step 4: Combine the results from Steps 2 and 3. The only restrictions on the domain come from the second term, where \( x = \pm 1 \) must be excluded. For all other real values of \( x \), the function is defined.
Step 5: Express the domain in interval notation, excluding \( x = \pm 1 \). The domain is \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).
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Domain of a Function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Understanding the domain is crucial for determining where the function can be evaluated without resulting in undefined expressions.
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3:51
Domain Restrictions of Composed Functions
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = 1/(x^2+1) - 1/(x^2-1), each term is a rational function. Analyzing rational functions involves identifying points where the denominator is zero, which directly affects the domain and behavior of the function.
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6:04Intro to Rational Functions
Finding Restrictions on the Domain
To find the domain of a function, one must identify any restrictions that arise from the function's structure. For the function f(x), we need to set the denominators of each term to zero and solve for x. The values that make the denominators zero are excluded from the domain, leading to a complete understanding of where the function is valid.
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3:51Domain Restrictions of Composed Functions
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