(a) Domain: (Enter your answer using interval notation.)
[tex]$\square$[/tex]
[tex]$[-5,7]$[/tex]
(b) Range: (Enter your answer using interval notation.)
[tex]$\square$[/tex]
[tex]$[-1,2]$[/tex]
(c) [tex]$f(1)$[/tex]
[tex]$\square$[/tex]
[tex]$-1$[/tex]
(d) all [tex]$x$[/tex] such that [tex]$f(x)=1$[/tex] (Enter your answers as a comma-separated list.)
[tex]$x=-3,-1,3,5$[/tex]
[tex]$\square$[/tex]
(e) all [tex]$x$[/tex] such that [tex]$f(x)\ \textgreater \ 1$[/tex] (Enter your answer using interval notation.)
[tex]$x \in[-1 I-3] \cup[3,5]$[/tex]
Answer (1)
The domain is [ − 5 , 7 ] .
The range is [ − 1 , 2 ] .
f ( 1 ) = − 1 .
f ( x ) = 1 for x = − 3 , − 1 , 3 , 5 .
1"> f ( x ) > 1 for x in [ − 3 , − 1 ] ∪ [ 3 , 5 ] .
Explanation
Understanding the Problem We are given a function and its graph (although not explicitly shown here, we assume we have access to it). We need to verify the given domain, range, the value of the function at a specific point, and the intervals where the function satisfies certain conditions.
Verifying the Domain The domain is the set of all possible x values for which the function is defined. From the given answer, the domain is [ − 5 , 7 ] . This means the function is defined for all x values between -5 and 7, inclusive. We assume this is correct based on the graph.
Verifying the Range The range is the set of all possible y values that the function can take. The given range is [ − 1 , 2 ] . This means the function's y values are between -1 and 2, inclusive. We assume this is correct based on the graph.
Verifying f(1) We are given that f ( 1 ) = − 1 . This means when x = 1 , the y value is -1. We assume this is correct based on the graph.
Verifying f(x) = 1 We are given that f ( x ) = 1 for x = − 3 , − 1 , 3 , 5 . This means the function's y value is 1 at these x values. We assume this is correct based on the graph.
Verifying f(x) > 1 We are given that 1"> f ( x ) > 1 for x in [ − 1 , − 3 ] ∪ [ 3 , 5 ] . This seems to have a typo. It should be x in [ − 3 , − 1 ] ∪ [ 3 , 5 ] . This means the function's y value is greater than 1 for x values in the interval ( − 3 , − 1 ) and ( 3 , 5 ) . The given answer x in [ − 1 I − 3 ] ∪ [ 3 , 5 ] is incorrect. The correct interval notation should be ( − 3 , − 1 ) in [ 3 , 5 ] .
Examples
Understanding the domain and range of a function is crucial in many real-world applications. For example, if we are modeling the height of a ball thrown in the air as a function of time, the domain would represent the time interval during which the ball is in the air, and the range would represent the possible heights the ball reaches. Similarly, in economics, the domain of a cost function might represent the number of units produced, and the range would represent the total cost of production. Knowing the domain and range helps us interpret the model and make realistic predictions.
