6.5 Comparing Linear Functions
1. The functions f(x) and g(x) are linear functions. The domain of each function is the set of all real numbers x such that 3 ≤ x ≤ 6. The table shows some ordered pairs belonging to f(x). The function g(x) is defined by the rule g(x) = 3x − 1. Find the initial value and the range of each function.
x / f(x)3 / 6
4 / 9
5 / 12
6 / 15
Answer: f(x)—Initial Value = 9
f(x)—Range: 6 ≤ f(x) ≤ 15
g(x)—Range: from the domain, take the smallest value of x and the largest value and plug it into the equation for g(x).
g(x) = 3x – 1, when x= 3 and x = 6
g(3) = 3(3) – 1 g(6) = 3(6) – 1
= 8 = 17
Therefore the range of g(x) would be: 8 ≤ g(x) ≤ 17
g(x)—Initial value is the smallest value of the range: 8
2. Compare the following functions.
• Hannah walked 3.5 hours at an average rate of 3 miles per hour. The function H(t) represents the distance Hannah walked in t hours.
• The graph below shows the distance Penelope walked, P(t) (in miles), as a function of time t (in hours).
Hannah: domain given in the problem as well as the rate.
For the range, take the smallest value of x and the largest and plug it in for x in the equation y = 3x.
Pennelope: Domain from the graph on the x-axis (where the line starts and ends)
Range: from the graph on the y-axis (where the line starts and ends)
Slope: from the graph apply rise over run
Hannah / PenelopeRate: 3mph / Rate:2mph
Domain: 0 ≤ x ≤ 3.5 / Domain: 0 ≤ x ≤ 3.5
Range:
0 ≤ h(x) ≤ 10.5 / Range:
0 ≤ p(x) ≤ 7
Hannah walks faster than Penelope, and has a larger range.
However, both Hannah and Penelope have the same domain.