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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 89
Chapter 3, Problem 89

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.
Graph of a function increasing from left to right, approaching but not touching the x-axis, with f(4) value missing.

Verified step by step guidance
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Step 1: Determine the domain of the function by looking at the x-values for which the graph is defined. Since the graph extends from negative infinity to positive 4 (and beyond), the domain includes all real numbers less than or equal to 4. Write the domain as \((-\infty, 4]\).
Step 2: Determine the range of the function by observing the y-values the graph takes. The graph approaches the x-axis (y=0) but never touches it, and it increases without bound as x increases. So, the range is all y-values greater than 0, written as \((0, \infty)\).
Step 3: Identify the x-intercepts by finding where the graph crosses the x-axis (where \(y=0\)). Since the graph approaches but never touches the x-axis, there are no x-intercepts.
Step 4: Identify the y-intercept by finding where the graph crosses the y-axis (where \(x=0\)). From the graph, observe the y-value at \(x=0\) and note it as the y-intercept.
Step 5: Find the missing function value \(f(4)\) by locating \(x=4\) on the graph and reading the corresponding y-value. This value is the output of the function at \(x=4\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. In a graph, it corresponds to the horizontal extent of the curve. Understanding the domain helps determine where the function exists and can be evaluated.
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Domain Restrictions of Composed Functions

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. On a graph, it is the vertical spread of the curve. Knowing the range is essential to understand the behavior and limits of the function's outputs.
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Intercepts and Asymptotes

Intercepts are points where the graph crosses the axes: x-intercepts occur where y=0, and y-intercepts where x=0. An asymptote is a line the graph approaches but never touches, indicating limits of the function. Recognizing these helps analyze function behavior and solve for missing values.
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Related Practice
Textbook Question

Express the given function h as a composition of two functions f and g so that h(x) = (f ○ g)(x). h(x) = (x2 + 2x - 1)4

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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.

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 + 4| +2

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Textbook Question

Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. h(x) = -|x+3|

Textbook Question

Use the graphs of f and g to solve Exercises 83–90.

Find the domain of ƒ/g.

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Textbook Question

Use the graphs of f and g to solve Exercises 83–90.

Graph f+g.