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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 2 - Functions and GraphsProblema 38
Capitolo 3, Problema 38

Find f+gf+g, f−gf-g, fgfg, and fg\(\frac{f}{g}\). Determine the domain for each function.
f(x)=5−x2f\(\left\)(x\(\right\))=5-x^2, g(x)=x2+4x−12g\(\left\)(x\(\right\))=x^2+4x-12

Guida verificata passo dopo passo
1
Step 1: To find \( (f+g)(x) \), add the functions \( f(x) \) and \( g(x) \). This means you will add \( 5 - x^2 \) and \( x^2 + 4x - 12 \). Combine like terms to simplify.
Step 2: To find \( (f-g)(x) \), subtract \( g(x) \) from \( f(x) \). This involves subtracting \( x^2 + 4x - 12 \) from \( 5 - x^2 \). Again, combine like terms to simplify.
Step 3: To find \( (fg)(x) \), multiply the functions \( f(x) \) and \( g(x) \). This requires distributing \( 5 - x^2 \) across \( x^2 + 4x - 12 \) and combining like terms.
Step 4: To find \( \left(\frac{f}{g}\right)(x) \), divide \( f(x) \) by \( g(x) \). This means writing \( \frac{5 - x^2}{x^2 + 4x - 12} \). Simplify if possible, and identify any restrictions on the domain where the denominator is zero.
Step 5: Determine the domain for each function. For \( f+g \) and \( f-g \), the domain is all real numbers. For \( fg \), the domain is also all real numbers. For \( \frac{f}{g} \), exclude values that make the denominator zero by solving \( x^2 + 4x - 12 = 0 \) to find the restricted values.

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Function Operations

Function operations involve combining two or more functions through addition, subtraction, multiplication, or division. For example, if f(x) and g(x) are two functions, their sum is defined as (f + g)(x) = f(x) + g(x). Understanding these operations is crucial for manipulating and analyzing functions in algebra.
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Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For instance, in rational functions, the domain excludes values that make the denominator zero. Identifying the domain is essential for understanding the behavior and limitations of a function.
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Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax^2 + bx + c. They have a parabolic graph and can have various properties such as vertex, axis of symmetry, and roots. Recognizing the characteristics of quadratic functions is important for solving equations and analyzing their graphs.
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