Textbook Question
Find f + g, f - g, fg, and f/g. f(x) = x2 + x + 1, g(x) = x2 -1
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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 82
Graph each linear function. 6x-5f(x) - 20 = 0
Verified step by step guidance1
Rewrite the given equation to isolate the function \( f(x) \). The original equation is \( 6x - 5f(x) - 20 = 0 \). Add \( 5f(x) \) and subtract \( 6x \) from both sides to get \( -5f(x) = -6x + 20 \).
Divide both sides of the equation by \( -5 \) to solve for \( f(x) \). This gives \( f(x) = \frac{-6x + 20}{-5} \).
Simplify the right-hand side by dividing each term separately: \( f(x) = \frac{-6x}{-5} + \frac{20}{-5} \), which simplifies to \( f(x) = \frac{6}{5}x - 4 \).
Identify the slope and y-intercept from the equation \( f(x) = \frac{6}{5}x - 4 \). The slope \( m = \frac{6}{5} \) and the y-intercept \( b = -4 \).
To graph the function, start by plotting the y-intercept at \( (0, -4) \). Then use the slope \( \frac{6}{5} \) to find another point by rising 6 units and running 5 units to the right. Draw a straight line through these points to complete the graph.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Functions
A linear function is a function whose graph is a straight line, typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. Understanding linear functions helps in identifying how changes in x affect the output f(x).
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Linear Inequalities
Rearranging Equations
Rearranging equations involves manipulating the given equation to isolate the dependent variable, usually f(x) or y, on one side. This step is essential to rewrite the function in slope-intercept form, making it easier to graph.
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Graphing Linear Equations
Graphing linear equations requires plotting points or using the slope and y-intercept to draw the line on the coordinate plane. Recognizing the slope and intercept from the equation allows for accurate and efficient graphing.
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Related Practice
Textbook Question
Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).
h(x) = 1/(2x-3)
Textbook Question
In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine the domain for each function. f(x) = 3x - 1, g(x) = x - 5
Textbook Question
Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = |x|+3
Textbook Question
In Exercises 77–92, use the graph to determine a. the function's range; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
Textbook Question
In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
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