Question

The graph shows the functions f(x), p(x), and g(x):

Answer 1

Answer with explanation:

• We are given a function f(x) as:

                            $f(x)=y=2+(1.5)^x$

• The function g(x) is given by:

The straight line g of x joins ordered pairs (1, 1) and (3,-3).

We know that the equation of a straight line passing through two points (a,b) and (c,d) is given by:

$y-b=\dfrac{d-b}{c-a}\times (x-a)$

Here we have:

(a,b)=(1,1) and (c,d)=(3,-3)

Hence,

$y-1=\dfrac{-3-1}{3-1}\times (x-1)\\\\\\y-1=\dfrac{-4}{2}\times (x-1)\\\\\\y-1=-2(x-1)\\\\y-1=-2x+2\\\\y=-2x+2+1\\\\y=-2x+3$

                             Hence,

                                  $g(x)=-2x+3$

• Similarly, p(x) is a straight line passing through (4,2) and (2,-1).

Hence,

$y-2=\dfrac{-1-2}{2-4}\times (x-4)\\\\\\y-2=\dfrac{-3}{-2}\times (x-4)\\\\\\y-2=\dfrac{3}{2}\times (x-4)\\\\\\y-2=\dfrac{3}{2}x-6\\\\\\y=\dfrac{3}{2}x-6+2\\\\\\y=\dfrac{3}{2}x-4$

               Hence,

                   $p(x)=\dfrac{3}{2}x-4$

PART A:

We are asked to find the solution to the pair of equations p(x) and f(x).

i.e. we are asked to find the value of 'x' such that:

                  p(x)=f(x)

i.e. the x-value of the point of intersection of the graph of p(x) and f(x).

As we could observe that the graph f(x) and p(x) do not intersect hence, we get NO SOLUTION.

PART B:

We have to find two solution for f(x).

i.e. we have to find the value of function f(x) corresponding to two values of x.

1.   x=0 then f(x)=2+(1.5)^0=2
2.   x=1 f(x)=2+1.5=3.5

PART C:

The solution of f(x)=g(x) is the x-value of the point of intersection of graph f(x) and g(x).

Hence,

The point is: (0,3)

Hence, the solution is: x=0

Answer 2

Answer:

Part A:

The solution to the pair of equations represented by p(x) and f(x)

is (2 , -1)

Part B:

(1 , 1) and (3 , -3) are the two solutions for f(x)

Part C:

The solution to the pair of equations represented by g(x) = f(x) is (0 , 3)

Step-by-step explanation:

* Lets study the three graphs

- g(x) is an exponential function where f(x) = 2 + (1.5)^x

- g(x) intersect the y-axis at point (0 , 3)

- g(x) intersected f(x) at point (0 , 3)

- f(x) is a linear function passing through point (3 , -3) , (1 , 1)

- The slop of f(x) = 1 - -3/1 - 3 = 4/-2 = -2

- f(x) intersect y-axis at point (0 , 3)

- f(x) = -2x + 3

- f(x) intersect x-axis at point (1.5 , 0)

- f(x) intersected p(x) at point (2 , -1)

- p(x) is a linear function passing through point (4 , 2) , (2 , -1)

- The slop of p(x) = -1 - 2/2 - 4 = -3/-2 = 3/2

- p(x) intersect y-axis at point (0 , -4)

- p(x) = 3/2 x - 4

- p(x) intersect x-axis at point (8/3 , 0)

* Now lets solve the problem

* Part A:

∵ p(x) meet f(x) at point (2 , -1)

∴ The solution to the pair of equations represented by p(x) and f(x)

   is (2 , -1)

* Part B:

∵ f(x) passing through (1 , 1) and (3 , -3)

∴ (1 , 1) and (3 , -3) are the two solutions for f(x)

∵ g(x) meet f(x) at point (0 , 3)

∴ The solution to the pair of equations represented by g(x) = f(x)

   is (0 , 3)