QuestionIdentify the function with two distinct real zeros: a. $f(x)=2 x+2$ , b. $f(x)=(x-2)^{2}$ , c. $f(x)=x^{3}+2$ , d. $f(x)=|x|-2$ .

Studdy Solution

STEP 1

Assumptions1. A real zero of a function is a real number that makes the function equal to zero. . Distinct real zeros mean different real numbers that make the function equal to zero.
3. We need to find a function that has exactly two distinct real zeros.

STEP 2

Let's start with option a. $f(x)=2x+2$ . We can find the zero of this function by setting it equal to zero and solving for x.
$2x+2=0$

STEP 3

Subtract2 from both sides of the equation to isolate the term with x.
$2x = -2$

STEP 4

Divide both sides of the equation by2 to solve for x.
$x = -1$ This function has only one real zero, so option a is not the correct answer.

STEP 5

Now, let's consider option b. $f(x)=(x-2)^{2}$ . Again, we find the zero of this function by setting it equal to zero and solving for x.
$(x-2)^{2}=0$

STEP 6

Take the square root of both sides of the equation to solve for x.
$x-2 =0$

STEP 7

Add2 to both sides of the equation to solve for x.
$x =2$ This function also has only one real zero (although it is a double root), so option b is not the correct answer.

STEP 8

Now, let's consider option c. $f(x)=x^{3}+2$ . Again, we find the zero of this function by setting it equal to zero and solving for x.
$x^{3}+2=0$

STEP 9

Subtract2 from both sides of the equation to isolate the term with x.
$x^{3} = -2$

STEP 10

Take the cube root of both sides of the equation to solve for x.
$x = \sqrt[3]{-2}$ This function has only one real zero, so option c is not the correct answer.

STEP 11

Finally, let's consider option d. $f(x)=|x|-$ . The absolute value function has two zeros if the number subtracted from the absolute value is positive. In this case, we are subtracting, so this function has two zeros.
The zeros of this function are x = - and x =, so option d is the correct answer.

Was this helpful?

QuestionIdentify the function with two distinct real zeros: a. $f(x)=2 x+2$ , b. $f(x)=(x-2)^{2}$ , c. $f(x)=x^{3}+2$ , d. $f(x)=|x|-2$ .

Studdy Solution

STEP 1

Assumptions1. A real zero of a function is a real number that makes the function equal to zero. . Distinct real zeros mean different real numbers that make the function equal to zero.
3. We need to find a function that has exactly two distinct real zeros.

STEP 2

Let's start with option a. $f(x)=2x+2$ . We can find the zero of this function by setting it equal to zero and solving for x.
$2x+2=0$

STEP 3

Subtract2 from both sides of the equation to isolate the term with x.
$2x = -2$

STEP 4

Divide both sides of the equation by2 to solve for x.
$x = -1$ This function has only one real zero, so option a is not the correct answer.

STEP 5

Now, let's consider option b. $f(x)=(x-2)^{2}$ . Again, we find the zero of this function by setting it equal to zero and solving for x.
$(x-2)^{2}=0$

STEP 6

Take the square root of both sides of the equation to solve for x.
$x-2 =0$

STEP 7

Add2 to both sides of the equation to solve for x.
$x =2$ This function also has only one real zero (although it is a double root), so option b is not the correct answer.

STEP 8

Now, let's consider option c. $f(x)=x^{3}+2$ . Again, we find the zero of this function by setting it equal to zero and solving for x.
$x^{3}+2=0$

STEP 9

Subtract2 from both sides of the equation to isolate the term with x.
$x^{3} = -2$

STEP 10

Take the cube root of both sides of the equation to solve for x.
$x = \sqrt[3]{-2}$ This function has only one real zero, so option c is not the correct answer.

STEP 11

Finally, let's consider option d. $f(x)=|x|-$ . The absolute value function has two zeros if the number subtracted from the absolute value is positive. In this case, we are subtracting, so this function has two zeros.
The zeros of this function are x = - and x =, so option d is the correct answer.

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.

QuestionIdentify the function with two distinct real zeros: a. f(x)=2x+2f(x)=2 x+2f(x)=2x+2, b. f(x)=(x−2)2f(x)=(x-2)^{2}f(x)=(x−2)2, c. f(x)=x3+2f(x)=x^{3}+2f(x)=x3+2, d. f(x)=∣x∣−2f(x)=|x|-2f(x)=∣x∣−2.

Studdy Solution

STEP 1

Assumptions1. A real zero of a function is a real number that makes the function equal to zero. . Distinct real zeros mean different real numbers that make the function equal to zero.3. We need to find a function that has exactly two distinct real zeros.

STEP 2

Let's start with option a. f(x)=2x+2f(x)=2x+2f(x)=2x+2. We can find the zero of this function by setting it equal to zero and solving for x.2x+2=02x+2=02x+2=0

STEP 3

Subtract2 from both sides of the equation to isolate the term with x.2x=−22x = -22x=−2

STEP 4

Divide both sides of the equation by2 to solve for x.x=−1x = -1x=−1This function has only one real zero, so option a is not the correct answer.

STEP 5

Now, let's consider option b. f(x)=(x−2)2f(x)=(x-2)^{2}f(x)=(x−2)2. Again, we find the zero of this function by setting it equal to zero and solving for x.(x−2)2=0(x-2)^{2}=0(x−2)2=0

STEP 6

Take the square root of both sides of the equation to solve for x.x−2=0x-2 =0x−2=0

STEP 7

Add2 to both sides of the equation to solve for x.x=2x =2x=2This function also has only one real zero (although it is a double root), so option b is not the correct answer.

STEP 8

Now, let's consider option c. f(x)=x3+2f(x)=x^{3}+2f(x)=x3+2. Again, we find the zero of this function by setting it equal to zero and solving for x.x3+2=0x^{3}+2=0x3+2=0

STEP 9

Subtract2 from both sides of the equation to isolate the term with x.x3=−2x^{3} = -2x3=−2

STEP 10

Take the cube root of both sides of the equation to solve for x.x=−23x = \sqrt[3]{-2}x=3−2​This function has only one real zero, so option c is not the correct answer.

STEP 11

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.

QuestionIdentify the function with two distinct real zeros: a. $f(x)=2 x+2$ , b. $f(x)=(x-2)^{2}$ , c. $f(x)=x^{3}+2$ , d. $f(x)=|x|-2$ .

Assumptions1. A real zero of a function is a real number that makes the function equal to zero. . Distinct real zeros mean different real numbers that make the function equal to zero.
3. We need to find a function that has exactly two distinct real zeros.

Let's start with option a. $f(x)=2x+2$ . We can find the zero of this function by setting it equal to zero and solving for x.
$2x+2=0$

Subtract2 from both sides of the equation to isolate the term with x.
$2x = -2$

Divide both sides of the equation by2 to solve for x.
$x = -1$ This function has only one real zero, so option a is not the correct answer.

Now, let's consider option b. $f(x)=(x-2)^{2}$ . Again, we find the zero of this function by setting it equal to zero and solving for x.
$(x-2)^{2}=0$

Take the square root of both sides of the equation to solve for x.
$x-2 =0$

Add2 to both sides of the equation to solve for x.
$x =2$ This function also has only one real zero (although it is a double root), so option b is not the correct answer.

Now, let's consider option c. $f(x)=x^{3}+2$ . Again, we find the zero of this function by setting it equal to zero and solving for x.
$x^{3}+2=0$

Subtract2 from both sides of the equation to isolate the term with x.
$x^{3} = -2$

Take the cube root of both sides of the equation to solve for x.
$x = \sqrt[3]{-2}$ This function has only one real zero, so option c is not the correct answer.