Question

100 POINTS HELP EXPERTS PLEAASE!
The graph shows the functions f(x), p(x), and g(x):

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)? (4 points)

Part B: Write any two solutions for f(x). (4 points)

Part C: What is the solution to the equation p(x) = g(x)? Justify your answer. (6 points)

Answer 1

Answer:

  A)  (-3, -2)

  B)  (-7, 3) and (-3, -2)

  C)  (4.074, 1.032)

Step-by-step explanation:

An ordered pair is a solution to an equation if it satisfies the equation — makes it true. The given points are solutions to the functions whose graphs pass through those points.

Part A.

The function p(x) is defined to pass through points (4, 1) and (-3, -2).

The function f(x) is defined to pass through points (-7, 3) and (-3, -2).

These function definitions have point (-3, -2) in common.

(-3, -2) is the solution to the equation p(x) = f(x).

Part B.

The function f(x) is defined to pass through points (-7, 3) and (-3, -2).

Two solutions to f(x) are (-7, 3) and (-3, -2).

We could identify other solutions, (1, -7) for example, but there is no need since the problem statement already gives us two solutions.

Part C.

The solution to the equation p(x) = g(x) can be read from the graph as approximately (4.074, 1.032). This is close to the point (4, 1) that is used to define p(x). With some refinement (iteration), we can show the irrational solution is closer to ...

  (4.07369423957, 1.03158324553)

Answer 2

Answer:

A)  (-3, -2)

B)  (1, -7) and (5, -12)

C)  (4, 1) to the nearest whole number

Step-by-step explanation:

Function g(x):

$g(x)=2(0.85)^x$

Function f(x) (straight line):

Given ordered pairs:

• Let (x₁, y₁) = (-7, 3)
• Let (x₂, y₂) = (-3, -2)

Calculate the slope of the straight line:

$\textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-2-3}{-3-(-7)}=-\dfrac{5}{4}$

Using the Point-slope form of linear equation:

$\implies y-y_1=m(x-x_1)$

$\implies y-3=-\dfrac{5}{4}(x-(-7))$

$\implies y=-\dfrac{5}{4}x-\dfrac{23}{4}$

$\implies f(x)=-\dfrac{5}{4}x-\dfrac{23}{4}$

Function p(x) (straight line):

Given ordered pairs:

• Let (x₁, y₁) = (4, 1)
• Let (x₂, y₂) = (-3, -2)

Calculate the slope of the straight line:

$\textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{-2-1}{-3-4}=\dfrac{3}{7}$

Using the Point-slope form of linear equation:

$\implies y-y_1=m(x-x_1)$

$\implies y-1=\dfrac{3}{7}(x-4)$

$\implies y=\dfrac{3}{7}x-\dfrac{5}{7}$

$\implies p(x)=\dfrac{3}{7}x-\dfrac{5}{7}$

Part A

We have been given two ordered pairs for function f(x) and function p(x).

One of those ordered pairs is the same for both functions.  

The solution to a pair of equations is their point(s) of intersection.  

Therefore, as both functions pass through (-3, -2), this is their point of intersection and therefore the solution.

Part B

The solutions for f(x) are any points on the line of the function f(x).  

To find any two points, substitute values of x into the found equation for f(x):

$\implies f(1)=-\dfrac{5}{4}(1)-\dfrac{23}{4}=-7$

$\implies f(5)=-\dfrac{5}{4}(5)-\dfrac{23}{4}=-12$

Therefore, two solutions are (1, -7) and (5, -12).

Part C

The solution to p(x) = g(x) is where the two graphs intersect.  From inspection of the graphs, p(x) intersects g(x) at approximately (4, 1).  

Therefore, the approximate solution to p(x) = g(x) is (4, 1).

To prove this, substitute x = 4 into the equations for p(x) and g(x):

$\implies p(4)=\dfrac{3}{7}(4)-\dfrac{5}{7}=1$

$\implies g(4)=2(0.85)^4=1.0440125=1.0\:(\sf nearest\:tenth)$

The actual solution to p(x) = g(x) is (4.074, 1.032) to three decimal places, which can be found by equating the functions and solving for x using a numerical method such as iteration.