Textbook Question
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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Table of contents
- 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
Chapter 3, Problem 30a
Evaluate each function at the given values of the independent variable and simplify. g(x) = x² - 10x - 3 a. g(-1)
Verified step by step guidance1
Substitute the given value of the independent variable, x = -1, into the function g(x) = x² - 10x - 3. This means replacing every occurrence of x in the function with -1.
The function becomes g(-1) = (-1)² - 10(-1) - 3. Simplify each term individually.
First, calculate (-1)². Recall that squaring a negative number results in a positive value, so (-1)² = 1.
Next, calculate -10(-1). Multiplying a negative number by another negative number results in a positive value, so -10(-1) = 10.
Finally, combine all the terms: g(-1) = 1 + 10 - 3. Simplify the expression by performing the addition and subtraction in order.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Evaluation
Function evaluation involves substituting a specific value for the independent variable in a function to find the corresponding output. In this case, to evaluate g(-1), we replace x in the function g(x) = x² - 10x - 3 with -1, allowing us to compute the value of the function at that point.
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Quadratic Functions
A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c, where a, b, and c are constants. The function g(x) = x² - 10x - 3 is a quadratic function, and its graph is a parabola that opens upwards, which is essential for understanding its behavior and properties.
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Simplification
Simplification in mathematics refers to the process of reducing an expression to its simplest form. After evaluating the function g(-1), it is important to simplify the resulting expression to make it easier to interpret and understand, ensuring that all like terms are combined and the expression is presented clearly.
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Related Practice
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