Answer :
Part A:
Given that the straight line p(x) joins the ordered pairs (0, 2) and (1, -5), thus the equation of the line joining ordered pairs (0, 2) and (1, -5) is given by
[tex] \frac{y-2}{x} = \frac{-5-2}{1} =-7 \\ \\ \Rightarrow y-2=-7x \\ \\ \Rightarrow y=-7x+2[/tex]
Thus, p(x) = -7x + 2
Given that the straight line f(x) joins the ordered pairs (4, 1) and (2, -3), thus the equation of the line joining ordered pairs (4, 1) and (2, -3) is given by
[tex] \frac{y-1}{x-4} = \frac{-3-1}{2-4} =\frac{-4}{-2}=2 \\ \\ \Rightarrow y-1=2(x-4)=2x-8 \\ \\ \Rightarrow y=2x-7[/tex]
Thus, f(x) = 2x - 7
The solution to the pair of equations represented by p(x) and f(x) is given by
p(x) = f(x)
⇒ -7x + 2 = 2x - 7
⇒ -7x - 2x = -7 - 2
⇒ -9x = -9
⇒ x = -9 / -9 = 1
Substituting for x into p(x), we have
p(1) = -7(1) + 2 = -7 + 2 = -5
Therefore, the solution to the pair of equations represented by p(x) and f(x) is (1, -5)
Part B:
From part A, we have that f(x) = 2x - 7
when x = -8
f(-8) = 2(-8) - 7 = -23
Thus, (-8, -23) is a solution to f(x).
When x = -10
f(-10) = 2(-10) - 7 = -27
Thus, (-10, -27) is a solution to f(x).
Therefore, two solutions of f(x) are (-8, -23) and (-10, -27).
Part C:
From part A, we have that p(x) = -7x + 2, given that [tex]g(x) = 1 + 1.5^x[/tex]
From the graphs of p(x) and g(x), we can see that the two graphs intersected at the point (0, 2).
Therefore, the solution to the equation p(x) = g(x) is (0, 2).
Given that the straight line p(x) joins the ordered pairs (0, 2) and (1, -5), thus the equation of the line joining ordered pairs (0, 2) and (1, -5) is given by
[tex] \frac{y-2}{x} = \frac{-5-2}{1} =-7 \\ \\ \Rightarrow y-2=-7x \\ \\ \Rightarrow y=-7x+2[/tex]
Thus, p(x) = -7x + 2
Given that the straight line f(x) joins the ordered pairs (4, 1) and (2, -3), thus the equation of the line joining ordered pairs (4, 1) and (2, -3) is given by
[tex] \frac{y-1}{x-4} = \frac{-3-1}{2-4} =\frac{-4}{-2}=2 \\ \\ \Rightarrow y-1=2(x-4)=2x-8 \\ \\ \Rightarrow y=2x-7[/tex]
Thus, f(x) = 2x - 7
The solution to the pair of equations represented by p(x) and f(x) is given by
p(x) = f(x)
⇒ -7x + 2 = 2x - 7
⇒ -7x - 2x = -7 - 2
⇒ -9x = -9
⇒ x = -9 / -9 = 1
Substituting for x into p(x), we have
p(1) = -7(1) + 2 = -7 + 2 = -5
Therefore, the solution to the pair of equations represented by p(x) and f(x) is (1, -5)
Part B:
From part A, we have that f(x) = 2x - 7
when x = -8
f(-8) = 2(-8) - 7 = -23
Thus, (-8, -23) is a solution to f(x).
When x = -10
f(-10) = 2(-10) - 7 = -27
Thus, (-10, -27) is a solution to f(x).
Therefore, two solutions of f(x) are (-8, -23) and (-10, -27).
Part C:
From part A, we have that p(x) = -7x + 2, given that [tex]g(x) = 1 + 1.5^x[/tex]
From the graphs of p(x) and g(x), we can see that the two graphs intersected at the point (0, 2).
Therefore, the solution to the equation p(x) = g(x) is (0, 2).