Domanda del libro di testo
In Exercises 31–32, the domain of each piecewise function is (-∞, ∞) (a) Graph each function. (b) Use the graph to determine the function's range.
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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 30c
Evaluate each function at the given values of the independent variable and simplify. g(x) = x² - 10x - 3 c. g(-x)
Guida verificata passo dopo passo1
Step 1: Understand the problem. You are given a function g(x) = x² + 2x + 3, and you are tasked with evaluating g(-x). This means substituting -x in place of x in the function.
Step 2: Substitute -x into the function. Replace every occurrence of x in g(x) with -x. The function becomes g(-x) = (-x)² + 2(-x) + 3.
Step 3: Simplify the first term (-x)². Recall that squaring a negative number results in a positive value, so (-x)² = x².
Step 4: Simplify the second term 2(-x). Multiply 2 by -x to get -2x.
Step 5: Combine all simplified terms. The function g(-x) simplifies to g(-x) = x² - 2x + 3.
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Function Evaluation
Function evaluation involves substituting a specific value for the independent variable in a function to determine its output. For example, if g(x) = x² - 10x - 3, evaluating g(-x) means replacing x with -x, resulting in g(-x) = (-x)² - 10(-x) - 3. This process is fundamental in understanding how functions behave under different inputs.
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Evaluating Composed Functions
Simplification of Expressions
Simplification of expressions refers to the process of reducing a mathematical expression to its simplest form. This often involves combining like terms, factoring, or applying algebraic identities. For instance, after evaluating g(-x), one would simplify the resulting expression to make it easier to interpret or use in further calculations.
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Quadratic Functions
Quadratic functions are polynomial functions of degree two, typically expressed in the form g(x) = ax² + bx + c, where a, b, and c are constants. Understanding the properties of quadratic functions, such as their parabolas' shape and vertex, is crucial for evaluating and simplifying expressions involving them. In the given question, both functions are quadratic, which influences how we evaluate and simplify them.
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06:36Solving Quadratic Equations Using The Quadratic Formula
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