Question

to solve the equation |x-9|=0.5x, kiana graphed the functions f(x)=|x-9| and g(x)=0.5x on the same set of coordinate axes. She then found that the graphs intersected at the points (6,3) and (18,9). Finally, she concluded that the solutions of the equation |x-9|=0.5x are x=6 amd x=18. which of the following reasons best justifies Kiana's conclusion.

Answer 1

Option: D is the correct answer.

F(6)=G(6) and F(18)=G(18)

Explanation:

We know that the point of intersection of the graph of two function f(x) and g(x) is the set of possible x values at which both the function have the same output values.

i.e. if the graph of two functions intersect at x=a than that means:

f(a)=g(a)

Hence, here the graph of F(x)=|x-9| and G(x)=0.5x intersect at x=6 and x=18 ,

this means that:

F(6)=G(6)

and F(18)=G(18)

Answer 2

The answer is the option D

`f(6)=g(6)` and
`f(18)=g(18)`

Explanation:

we have

`f\left(x\right)=\left|x-9\right|`

`g(x)=0.5x`

we know that

The solution of the system of equations is the intersection point both graphs

Using a graphing tool

see the attached figure

The solution are the points
`(6,3)` and
`(18,9)`

therefore

The solution of the equation
`\left|x-9\right|=0.5x` are

`x=6, x=18`

so

For
`x=6`

`f\left(6\right)=\left|6-9\right|=3`

`g(6)=0.5(6)=3`

`f(6)=g(6)`

For
`x=18`

`f\left(18\right)=\left|18-9\right|=9`

`g(18)=0.5(18)=9`

`f(18)=g(18)`

therefore

`f(6)=g(6)` and
`f(18)=g(18)` because the intersection points are common points for both graphs