QuestionCan different functions share the same $x$ -intercepts $(-5,0)$ , $(2,0)$ , $(6,0)$ , domain $-5 \leq x \leq 7$ , and range $-4 \leq y \leq 10$ ? Provide examples.

Studdy Solution

STEP 1

Assumptions1. The x-intercepts are $(-5,0),(,0)$ , and $(6,0)$ . The domain is $-5 \leq x \leq7$
3. The range is $-4 \leq y \leq10$

STEP 2

First, let's consider the simplest function that meets these conditions, a linear function. The x-intercepts indicate that the function should be zero at x = -5,2, and6. A possible function could be $f(x) = (x+5)(x-2)(x-6)$ This function is zero at the given x-intercepts.

STEP 3

Now, let's check if this function meets the domain and range requirements. The domain is already satisfied because this function is defined for all real numbers.

STEP 4

To check the range, we need to find the maximum and minimum values of the function within the given domain.

STEP 5

The minimum and maximum values of a polynomial function occur at its critical points, which are the points where the derivative is zero or undefined, and at the endpoints of the domain.

STEP 6

First, let's find the derivative of the function.
$f'(x) = (x+5)(x-2)(x-6)' = (x+5)(x-2)'(x-6) + (x+5)'(x-2)(x-6) + (x+5)(x-2)(x-6)'$

STEP 7

implify the derivative.
$f'(x) =3x^2 -18x +20$

STEP 8

Find the critical points by setting the derivative equal to zero and solving for x.
$3x^2 -18x +20 =0$

STEP 9

olve the quadratic equation to find the critical points.
$x = \frac{18 \pm \sqrt{(-18)^2 -4320}}{2*3}$

STEP 10

Calculate the values of x.
$x = \frac{18 \pm \sqrt{324 -240}}{6}$ $x = \frac{18 \pm \sqrt{84}}{6}$ $x =3 \pm \sqrt{7}$

STEP 11

The critical points are $x =3 - \sqrt{7}$ and $x =3 + \sqrt{7}$ . Now, let's evaluate the function at these points and at the endpoints of the domain to find the minimum and maximum values.

STEP 12

Evaluate the function at $x = - \sqrt{7}$ , $x = + \sqrt{7}$ , $x = -5$ , and $x =7$ .
$f( - \sqrt{7}) = ( - \sqrt{7} +5)( - \sqrt{7} -2)( - \sqrt{7} -6)$ $f( + \sqrt{7}) = ( + \sqrt{7} +5)( + \sqrt{7} -2)( + \sqrt{7} -6)$ $f(-5) = (-5 +5)(-5 -2)(-5 -6)$ $f(7) = (7 +5)(7 -2)(7 -6)$

STEP 13

Calculate the values of the function.
$f(3 - \sqrt{7}) = -$ $f(3 + \sqrt{7}) =10$ $f(-5) =0$ $f(7) =24$

STEP 14

The minimum value of the function within the given domain is -4, and the maximum value is24. Therefore, the range of the function is $-4 \leq y \leq24$ , which does not meet the given range requirement.

STEP 15

Now, let's try a different function. We can modify the function by adding a constant to it. Let's try the function $g(x) = (x+5)(x-2)(x-) -14$

STEP 16

This function is also zero at the given x-intercepts, and its range is shifted down by14 units compared to the previous function.

STEP 17

Now, let's check if this function meets the domain and range requirements. The domain is already satisfied because this function is defined for all real numbers.

STEP 18

To check the range, we need to find the maximum and minimum values of the function within the given domain.

STEP 19

The minimum and maximum values of the function occur at the same x-values as for the previous function, because the derivative of this function is the same as the derivative of the previous function.

STEP 20

Evaluate the function at $x =3 - \sqrt{7}$ , $x =3 + \sqrt{7}$ , $x = -5$ , and $x =7$ .
$g(3 - \sqrt{7}) = (3 - \sqrt{7} +5)(3 - \sqrt{7} -)(3 - \sqrt{7} -6) -14$ $g(3 + \sqrt{7}) = (3 + \sqrt{7} +5)(3 + \sqrt{7} -)(3 + \sqrt{7} -6) -14$ $g(-5) = (-5 +5)(-5 -)(-5 -6) -14$ $g(7) = (7 +5)(7 -)(7 -6) -14$

STEP 21

Calculate the values of the function.
$g(3 - \sqrt{7}) = -18$ $g(3 + \sqrt{7}) = -4$ $g(-5) = -14$ $g(7) =10$

STEP 22

The minimum value of the function within the given domain is -18, and the maximum value is10. Therefore, the range of the function is $-18 \leq y \leq10$ , which does not meet the given range requirement.

STEP 23

From these examples, we can see that it is not possible to find a function that meets all the given conditions. Therefore, Rhianna's claim is incorrect.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms