QuestionRhianna claims multiple functions can share $x$ -intercepts, domain, and range.
a. With $x$ -intercepts at $(-5,0),(2,0),(6,0)$ , domain $-5 \leq x \leq 7$ , and range $-4 \leq y \leq 10$ , can multiple functions exist?
b. With $x$ -intercepts at $(-4,0)$ and $(2,0)$ , domain as all real numbers, and range $y \geq -8$ , can multiple functions exist?

Studdy Solution

STEP 1

Assumptions1. The $x$ -intercepts are points where the function crosses or touches the $x$ -axis, i.e., where $y =0$ . . The domain of a function is the set of all possible $x$ -values.
3. The range of a function is the set of all possible $y$ -values.
4. We are looking for functions that satisfy the given conditions for $x$ -intercepts, domain, and range.

STEP 2

For part a, we have $x$ -intercepts at $(-5,0),(2,0)$ , and $(6,0)$ , the domain is $-5 \leq x \leq7$ , and the range is $-4 \leq y \leq10$ .

STEP 3

We can start by drawing a simple linear function that passes through the given $x$ -intercepts. A possible function is $f(x) = x +5$ for $-5 \leq x \leq2$ and $f(x) = -x +6$ for $2 < x \leq6$ and $f(x) = x -6$ for $6 < x \leq7$ . This function satisfies the given conditions.

STEP 4

We can also draw a quadratic function that passes through the given $x$ -intercepts. A possible function is $f(x) = (x+)(x-2)(x-6)$ , which also satisfies the given conditions.

STEP 5

Therefore, there are indeed multiple functions that have the same $x$ -intercepts, domain, and range. Rhianna is correct.

STEP 6

For part b, we have $x$ -intercepts at $(-4,0)$ and $(2,0)$ , the domain is all real numbers, and the range is $y \geq -8$ .

STEP 7

We can start by drawing a simple linear function that passes through the given $x$ -intercepts. A possible function is $f(x) = x +4$ for $-4 \leq x \leq2$ and $f(x) = -x +2$ for $2 < x \leq4$ . This function satisfies the given conditions.

STEP 8

We can also draw a quadratic function that passes through the given $x$ -intercepts. A possible function is $f(x) = (x+4)(x-2)$ , which also satisfies the given conditions.

STEP 9

Therefore, there are indeed multiple functions that have the same $x$ -intercepts, domain, and range. Rhianna is correct.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

For part a, we have $x$ -intercepts at $(-5,0),(2,0)$ , and $(6,0)$ , the domain is $-5 \leq x \leq7$ , and the range is $-4 \leq y \leq10$ .

STEP 3

We can start by drawing a simple linear function that passes through the given $x$ -intercepts. A possible function is $f(x) = x +5$ for $-5 \leq x \leq2$ and $f(x) = -x +6$ for $2 < x \leq6$ and $f(x) = x -6$ for $6 < x \leq7$ . This function satisfies the given conditions.

STEP 4

We can also draw a quadratic function that passes through the given $x$ -intercepts. A possible function is $f(x) = (x+)(x-2)(x-6)$ , which also satisfies the given conditions.

STEP 5

Therefore, there are indeed multiple functions that have the same $x$ -intercepts, domain, and range. Rhianna is correct.

STEP 6

For part b, we have $x$ -intercepts at $(-4,0)$ and $(2,0)$ , the domain is all real numbers, and the range is $y \geq -8$ .

STEP 7

We can start by drawing a simple linear function that passes through the given $x$ -intercepts. A possible function is $f(x) = x +4$ for $-4 \leq x \leq2$ and $f(x) = -x +2$ for $2 < x \leq4$ . This function satisfies the given conditions.

STEP 8

We can also draw a quadratic function that passes through the given $x$ -intercepts. A possible function is $f(x) = (x+4)(x-2)$ , which also satisfies the given conditions.

STEP 9

Therefore, there are indeed multiple functions that have the same $x$ -intercepts, domain, and range. Rhianna is correct.

Get Studdy

Studdy solves anything!

Start learning now

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

Studdy Solution

STEP 1

STEP 2

For part a, we have xxx-intercepts at (−5,0),(2,0)(-5,0),(2,0)(−5,0),(2,0), and (6,0)(6,0)(6,0), the domain is −5≤x≤7-5 \leq x \leq7−5≤x≤7, and the range is −4≤y≤10-4 \leq y \leq10−4≤y≤10.

STEP 3

STEP 4

We can also draw a quadratic function that passes through the given xxx-intercepts. A possible function is f(x)=(x+)(x−2)(x−6)f(x) = (x+)(x-2)(x-6)f(x)=(x+)(x−2)(x−6), which also satisfies the given conditions.

STEP 5

Therefore, there are indeed multiple functions that have the same xxx-intercepts, domain, and range. Rhianna is correct.

STEP 6

For part b, we have xxx-intercepts at (−4,0)(-4,0)(−4,0) and (2,0)(2,0)(2,0), the domain is all real numbers, and the range is y≥−8y \geq -8y≥−8.

STEP 7

STEP 8

We can also draw a quadratic function that passes through the given xxx-intercepts. A possible function is f(x)=(x+4)(x−2)f(x) = (x+4)(x-2)f(x)=(x+4)(x−2), which also satisfies the given conditions.

STEP 9

Therefore, there are indeed multiple functions that have the same xxx-intercepts, domain, and range. Rhianna is correct.

Get Studdy

Studdy solves anything!

Start learning now

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

For part a, we have $x$ -intercepts at $(-5,0),(2,0)$ , and $(6,0)$ , the domain is $-5 \leq x \leq7$ , and the range is $-4 \leq y \leq10$ .

We can also draw a quadratic function that passes through the given $x$ -intercepts. A possible function is $f(x) = (x+)(x-2)(x-6)$ , which also satisfies the given conditions.

Therefore, there are indeed multiple functions that have the same $x$ -intercepts, domain, and range. Rhianna is correct.

For part b, we have $x$ -intercepts at $(-4,0)$ and $(2,0)$ , the domain is all real numbers, and the range is $y \geq -8$ .

We can also draw a quadratic function that passes through the given $x$ -intercepts. A possible function is $f(x) = (x+4)(x-2)$ , which also satisfies the given conditions.

Therefore, there are indeed multiple functions that have the same $x$ -intercepts, domain, and range. Rhianna is correct.