Question

Graph of function g of x is y is equal to 2 multiplied by 0.85 to the power of x. The straight line f of x joins ordered pairs minus 7, 3 and minus 3, minus 2 and is extended on both sides. The straight line p of x joins the ordered pairs 4, 1 and minus 3, minus 2 and is extended on both sides.

Part A: What is the solution to the pair of equations represented by p(x) and f(x)? (3 points)

Part B: Write any two solutions for f(x). (3 points)

Part C: What is the solution to the equation p(x) = g(x)? Justify your answer. (4 points

Answer 1

p(x) = (3/7)x = 4/7
f(x) = .5x - .5

A. Solution to p(x) and f(x) is (15,7)
B. f(x) = .5x-.5
C (2.02,1.44)

Answer 2

i)
`g(x)=2(0.8)^(x)`

ii) the straight line joining (-7, 3) and (-3, -2), has slope


`m= (y_2-y_1)/(x_2-x_1)= (-2-3)/(-3-(-7))= (-5)/(4)`, thus the equation of the line can be written as:

y-3=(-5/4)(x+7)

y-3=(-5/4)x-35/4
y=(-5/4)x-35/4+3=(-5/4)x-35/4+12/4=(-5/4)x-23/4

finally, we can write the linear function as :

f(x)=(-5/4)x-23/4

iii) line p joins the points (4, 1), (-3, -2)

we follow the same steps as in ii) to write the linear function p(x):

the slope = (-2-1)/(-3-4)=-3/-7=3/7

the equation of this line is :

y-1=(3/7)(x-4)
y-1=(3/7)x-12/7
y=(3/7)x-12/7+1=(3/7)x-12/7+7/7=(3/7)x-5/7

thus the function is

p(x)=(3/7)x-5/7


Part A:

the solution to the pair of equations y=p(x) and y=f(x) is found by equalizing y:


p(x)=f(x)

(3/7)x-5/7=(-5/4)x-23/4

(3/7)x+(5/4)x=5/7-23/4

(12+35)/28 x = (20-161)/28

47x=-141
x=-3,

substitute x=-3 in p(x) (or f(x), it is the same) to find y:

y=(3/7)x-5/7=(3/7)(-3)-5/7=-9/7 -5/7 = -14/7 = -2


thus the solution is (-3, -2)


Part B.

the solution/s of f(x) are those x for which f(x)=0,

so we solve:

f(x)=(-5/4)x-23/4 =0

(-5/4)x-23/4 =0

(-5/4)x=23/4

-5x=23

x=-23/5

THERE ARE NO 2 SOLUTIONS (or better, roots) of f(x), there is only 1.

C:

p(x)=g(x)


`2(0.8)^(x)=(3/7)x-5/7`

p(x) is an exponential function with the base of the exponent < 1, so it is a decreasing function. W also know that the domain is all the real numbers.

g(x) is an increasing function, the graph of it is a line that increases, because the slope = 3/7 is positive.

We can way for sure that there is a unique solution, but to find it we can only use a graphing device,

like desmos.org, used in the attached picture, or another.

graphing p(x) and g(x), we see that the intersection is at (3.707, 0.875)

So the solution is x= 3.707