The table shows three unique, discrete functions:
| x | f(x) | g(x) | h(x) |
| --- | ---- | ----- | ---- |
| -1 | 2 | 2 | 2 |
| 0 | 3 | 3 | 3 |
| 3 | 2 | 0 | 4 |
| 8 | -3 | -25 | 5 |
| 15 | -10 | -204 | 6 |
Which statements can be used to compare the characteristics of the functions? Select three options.
A. [tex]$f(x)$[/tex] has the greatest maximum.
B. [tex]$h(x)$[/tex] has the greatest [tex]$x$[/tex]-intercept.
C. [tex]$g(x)$[/tex] has the smallest minimum value.
D. All three functions share the same domain.
E. All three functions share the same [tex]$y$[/tex]-intercept.
Answer (2)
Determine the maximum values of the functions and find that h ( x ) has the greatest maximum.
Determine the minimum values of the functions and find that g ( x ) has the smallest minimum value.
Compare the domains of the functions and find that they are the same.
Compare the y-intercepts of the functions and find that they are the same.
The correct statements are: g ( x ) has the smallest minimum value, all three functions share the same domain, and all three functions share the same y -intercept. g ( x ) has the smallest minimum value, same domain, same y-intercept.
Explanation
Understanding the Problem We are given a table of values for three discrete functions: f ( x ) , g ( x ) , and h ( x ) . Our goal is to identify three correct statements comparing characteristics of these functions from the given options.
Analyzing the Statements Let's analyze each statement:
' f ( x ) has the greatest maximum.'
The maximum value of f ( x ) is 3 (at x = 0 ).
The maximum value of g ( x ) is 3 (at x = 0 ).
The maximum value of h ( x ) is 6 (at x = 15 ).
Therefore, h ( x ) has the greatest maximum, so this statement is false.
' h ( x ) has the greatest x -intercept.'
To find the x -intercept, we need to find the value of x when the function equals zero. Since we only have discrete points, we can approximate the x-intercept by finding where the function changes sign. However, none of the functions change sign from positive to negative. We can try to approximate the x-intercept by linear interpolation between the points. For h ( x ) , we can consider the points ( 3 , 4 ) and ( 8 , 5 ) . The equation of the line passing through these points is: y − 4 = 8 − 3 5 − 4 ( x − 3 ) y − 4 = 5 1 ( x − 3 ) To find the x-intercept, set y = 0 :
− 4 = 5 1 ( x − 3 ) − 20 = x − 3 x = − 17 Since we don't have enough information to accurately determine the x-intercepts of f ( x ) and g ( x ) , and we found that the x-intercept of h ( x ) is -17 (approximated), we cannot definitively say that h ( x ) has the greatest x-intercept. However, since the other options are more clearly correct or incorrect, we will re-evaluate this statement later.
' g ( x ) has the smallest minimum value.'
The minimum value of f ( x ) is -10 (at x = 15 ).
The minimum value of g ( x ) is -204 (at x = 15 ).
The minimum value of h ( x ) is 2 (at x = − 1 ).
Therefore, g ( x ) has the smallest minimum value, so this statement is true.
'All three functions share the same domain.'
The domain of each function is the set of x values for which we have data. In this case, the domain for all three functions is { − 1 , 0 , 3 , 8 , 15 } .
Therefore, all three functions share the same domain, so this statement is true.
'All three functions share the same y -intercept.'
The y -intercept is the value of the function when x = 0 . From the table, f ( 0 ) = 3 , g ( 0 ) = 3 , and h ( 0 ) = 3 .
Therefore, all three functions share the same y -intercept, so this statement is true.
Identifying Correct Statements Based on our analysis, the following statements are true:
g ( x ) has the smallest minimum value.
All three functions share the same domain.
All three functions share the same y -intercept.
Final Answer Therefore, the three correct statements are:
g ( x ) has the smallest minimum value.
All three functions share the same domain.
All three functions share the same y -intercept.
Examples
Understanding the characteristics of different functions, such as their maximum and minimum values, intercepts, and domains, is crucial in various real-world applications. For example, in business, you might use functions to model sales trends. Knowing the maximum sales value helps in planning inventory, while the minimum value helps in setting safety stock levels. The domain of the function represents the time period under consideration, and the y-intercept indicates the starting sales value. Comparing these characteristics across different products or markets allows for informed decision-making and resource allocation.
The true statements are that g ( x ) has the smallest minimum value, all three functions share the same domain, and they all have the same y-intercept.
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