Determine the Domain of the Function Using Graph Analysis

Math Problem Statement

Use the graph of

yequals=f(x)

to answer the questions. Complete parts a through

-1717y

font size decreased by 2 left parenthesis negative 4 comma negative 3 right parenthesis(−4,−3)

font size decreased by 2 left parenthesis negative 2 comma 3 right parenthesis(−2,3)

font size decreased by 2 left parenthesis 2 comma negative 5 right parenthesis(2,−5)

font size decreased by 2 left parenthesis 4 comma negative 3 right parenthesis(4,−3)

font size decreased by 2 left parenthesis 5 comma negative 1 right parenthesis(5,−1)

font size decreased by 2 left parenthesis 8 comma negative 1 right parenthesis(8,−1)

font size decreased by 2 left parenthesis 9 comma 0 right parenthesis(9,0)

font size decreased by 2 left parenthesis 11 comma 2 right parenthesis(11,2)

font size decreased by 2 left parenthesis 13 comma 0 right parenthesis(13,0)

font size decreased by 2 left parenthesis 0 comma 0 right parenthesis(0,0)

font size decreased by 2 left parenthesis negative seven halves comma 0 right parenthesis−72,0

y equals f left parenthesis x right parenthesisy=f(x)

x6-6

A coordinate system has a horizontal x-axis labeled from negative 17 to 17 in increments of 1 and a vertical y-axis from negative 6 to 6 in increments of 1. The graph contains a curve labeled as y=f(x). From left to right, a curve start at open point (negative 4,negative 3) and rises to reach the point (negative 2,3), passing through the point (negative seven halves,0). From the point (negative 2,3) the curve falls to reach the point (2, negative 5), passing through the point (0,0). From point (2,negative 5) the curve rises to reach the point (5,negative 1), passing through the point (4,negative 3). From point (5,negative 1) to (8,negative 1) the curve does not rise or fall. From the point (8,negative 1) the curve rises to reach the point (11,2), passing through the point (9,0) and then falls; ending with the closed point (13,0).

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Question content area right

Part 1

a. What is the domain of f?

The domain is

enter your response here.

(Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

Solution

To find the domain of the function $f(x)$ , we need to determine the set of all $x$ -values for which the function $f(x)$ is defined.

Looking at the provided points and the description of the curve:

The curve starts at $x = -4$ with an open point at $(-4, -3)$ . This suggests that $x = -4$ is not included in the domain.
The curve continues through various points and ends at $x = 13$ with a closed point at $(13, 0)$ , indicating that $x = 13$ is included in the domain.

Since the curve is continuous from $x = -4$ (excluding $-4$ ) to $x = 13$ (including 13), the domain of $f(x)$ is: $(-4, 13]$

This is the interval notation for the domain of $f(x)$ .

Would you like details on how to answer the remaining parts? Here are some related questions:

What are the $x$ -intercepts of the function $f(x)$ ?
What is the range of $f(x)$ ?
Identify the local maxima and minima of $f(x)$ .
On which intervals is $f(x)$ increasing or decreasing?
Does $f(x)$ have any points of discontinuity?

Tip: When interpreting a graph, look for open and closed points as they often determine the inclusivity of domain and range boundaries.

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Math Problem Analysis

Mathematical Concepts

Graph Analysis
Domain and Range
Interval Notation

Formulas

Theorems

Domain and Range of Functions

Suitable Grade Level

Grades 9-12

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