Question

Rihanna says she can draw different functions that have the same x-intercepts and the same domain and range. Her teammates say, "No that's impossible." But she insists, "We just need to sketch some graphs."
1. x-int. are (-5,0), (2,0), and (6,0), domain is -5 Is there more than 1 possible function?
2. What if the x-intercepts are (-4,0) and (2,0), domain is all real #'s and the range is y>-8? Is there more than one possible function?

Answer 1

Answer with explanation:

1. X-Intercepts

      = (-5,0), (2,0), and (6,0)

It means the function cuts the x axis at three points, that is at points ,-5, 2 and 6.

Means the function is cubic.

Now, the function can be written as

  f(x)=(x+5)(x-2)(x-6)

 Now if we m ultiply the above function by any of the real numbers suppose ,k , the Zeroes will be same as well as Domain and range but the function will appear different.

f(x)=k×[(x+5)(x-2)(x-6)], where k is any real number.

2.

The function cuts x axis at two points that is at , (-4,0) and (2,0).

g(x)=(x+4)(x-2)

It is a quadratic function.

If you will multiply this function by any real number suppose , m , the function becomes different from given function but it's root will be same.

g(x)=m×(x+4)(x-2), where m is any real number.

So, there are many more functions in both the cases for distinct value of m and k.

Answer 2

1) Yes, there are many functions that have those x-intercepts with the same domain.

2) Yes, there are many functions with those x-intercpets, domain and range.

In fact, you might draw as many functions as you want that meet the specifications of both statements.