An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Determine the maximum and minimum values, x-intercepts, y-intercepts, and domains of each function from the table.
Evaluate each statement based on the characteristics found.
Identify the three correct statements.
The three 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.
Explanation
Understanding the Problem We are given a table of values for three discrete functions: f ( x ) , g ( x ) , and h ( x ) . We need to compare their characteristics based on the given data.
Analyzing the Table Let's analyze the table to determine the maximum and minimum values, x-intercepts, y-intercepts, and domains of each function.
Determining Function Characteristics
Maximum Values:
f ( x ) : The maximum value is 3 (at x = 0 ).
g ( x ) : The maximum value is 3 (at x = 0 ).
h ( x ) : The maximum value is 6 (at x = 15 ).
Minimum Values:
f ( x ) : The minimum value is -10 (at x = 15 ).
g ( x ) : The minimum value is -204 (at x = 15 ).
h ( x ) : The minimum value is 2 (at x = − 1 ).
X-Intercepts:
f ( x ) : The function value changes sign between x = 3 and x = 8 . We can't determine the exact x-intercept from the table, but we know it lies in the interval ( 3 , 8 ) .
g ( x ) : The function value is 0 at x = 3 , so the x-intercept is 3.
h ( x ) : The function value changes sign between x = 3 and x = 8 . We can't determine the exact x-intercept from the table, but we know it lies in the interval ( 3 , 8 ) .
Y-Intercepts:
f ( x ) : The y-intercept (value at x = 0 ) is 3.
g ( x ) : The y-intercept (value at x = 0 ) is 3.
h ( x ) : The y-intercept (value at x = 0 ) is 3.
Domains:
All three functions have the same domain: { − 1 , 0 , 3 , 8 , 15 } .
Evaluating the Statements Now, let's evaluate the given statements:
" f ( x ) has the greatest maximum." This is false because h ( x ) has a greater maximum (6) than f ( x ) (3).
" h ( x ) has the greatest x -intercept." This is false. g ( x ) has an x-intercept of 3. We know that the x-intercepts of f ( x ) and h ( x ) lie in the interval ( 3 , 8 ) , so we cannot say that h ( x ) has the greatest x-intercept.
" g ( x ) has the smallest minimum value." This is true because -204 is the smallest among -10, -204, and 2.
"All three functions share the same domain." This is true, as we determined the domain for each function is { − 1 , 0 , 3 , 8 , 15 } .
"All three functions share the same y -intercept." This is true, as all three functions have a y-intercept of 3.
Identifying Correct Statements 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 functions and their properties, such as maximum and minimum values, intercepts, and domains, is crucial in many real-world applications. For example, in economics, we might analyze cost functions to determine the minimum cost or revenue functions to find the maximum revenue. In physics, we can use functions to model the trajectory of a projectile and find its maximum height or range. In computer science, functions are used to model algorithms and analyze their efficiency. By understanding these concepts, we can make informed decisions and solve complex problems in various fields.
