Math / Algebra

QuestionEvaluating Func Given the function $f(x)=\sqrt{-4 x+4}$ . Calculate Note: Round the answer to two decimal places as I enter DNE $\begin{array}{l} f(-2)= \\ f(-1)= \\ \hline f(0)= \\ f(1)= \\ f(2)= \end{array}$

Studdy Solution

STEP 1

What is this asking? We need to plug different values into a function and see what pops out!
Specifically, we're given $f(x)$ and need to find $f(-2)$ , $f(-1)$ , $f(0)$ , $f(1)$ , and $f(2)$ . Watch out! Square roots can be tricky!
We can't take the square root of a negative number.

STEP 2

1. Evaluate $f(-2)$
2. Evaluate $f(-1)$
3. Evaluate $f(0)$
4. Evaluate $f(1)$
5. Evaluate $f(2)$

STEP 3

Let's kick things off with $x = -2$ !
We're going to substitute $-2$ for $x$ in our function $f(x) = \sqrt{-4x + 4}$ .
Remember, whatever we do on one side of the equation, we do on the other.

STEP 4

$f(-2) = \sqrt{-4 \cdot (-2) + 4}$ Multiplying $-4$ by $-2$ gives us positive $8$ .

STEP 5

$f(-2) = \sqrt{8 + 4}$ Adding $8$ and $4$ gives us $12$ .

STEP 6

$f(-2) = \sqrt{12}$ We can simplify $\sqrt{12}$ to $2\sqrt{3}$ .

STEP 7

$f(-2) \approx 3.46$ Rounding to two decimal places, we get 3.46!

STEP 8

Now, let's tackle $x = -1$ !
Substituting $-1$ for $x$ in $f(x)$ :

STEP 9

$f(-1) = \sqrt{-4 \cdot (-1) + 4}$ $-4$ times $-1$ is $4$ .

STEP 10

$f(-1) = \sqrt{4 + 4}$ $4$ plus $4$ is $8$ .

STEP 11

$f(-1) = \sqrt{8}$ Simplifying $\sqrt{8}$ gives us $2\sqrt{2}$ .

STEP 12

$f(-1) \approx 2.83$ Rounding to two decimal places, we get 2.83!

STEP 13

Time for $x = 0$ !
Substituting $0$ for $x$ in $f(x)$ :

STEP 14

$f(0) = \sqrt{-4 \cdot 0 + 4}$ Anything multiplied by $0$ is $0$ .

STEP 15

$f(0) = \sqrt{0 + 4}$ $0$ plus $4$ is $4$ .

STEP 16

$f(0) = \sqrt{4}$ The square root of $4$ is 2!

STEP 17

On to $x = 1$ !
Substituting $1$ for $x$ in $f(x)$ :

STEP 18

$f(1) = \sqrt{-4 \cdot 1 + 4}$ $-4$ times $1$ is $-4$ .

STEP 19

$f(1) = \sqrt{-4 + 4}$ $-4$ plus $4$ is $0$ .

STEP 20

$f(1) = \sqrt{0}$ The square root of $0$ is 0!

STEP 21

Lastly, let's conquer $x = 2$ !
Substituting $2$ for $x$ in $f(x)$ :

STEP 22

$f(2) = \sqrt{-4 \cdot 2 + 4}$ $-4$ times $2$ is $-8$ .

STEP 23

$f(2) = \sqrt{-8 + 4}$ $-8$ plus $4$ is $-4$ .

STEP 24

$f(2) = \sqrt{-4}$ Uh oh!
We can't take the square root of a negative number!

STEP 25

$f(-2) \approx 3.46$ $f(-1) \approx 2.83$ $f(0) = 2$ $f(1) = 0$ $f(2)$ is undefined (DNE)

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Studdy Solution

STEP 1

STEP 2

1. Evaluate f(−2)f(-2)f(−2)2. Evaluate f(−1)f(-1)f(−1)3. Evaluate f(0)f(0)f(0)4. Evaluate f(1)f(1)f(1)5. Evaluate f(2)f(2)f(2)

STEP 3

Let's **kick things off** with x=−2x = -2x=−2!We're going to substitute −2-2−2 for xxx in our function f(x)=−4x+4f(x) = \sqrt{-4x + 4}f(x)=−4x+4​.Remember, whatever we do on one side of the equation, we do on the other.

STEP 4

f(−2)=−4⋅(−2)+4f(-2) = \sqrt{-4 \cdot (-2) + 4}f(−2)=−4⋅(−2)+4​ Multiplying −4-4−4 by −2-2−2 gives us **positive** 888.

STEP 5

f(−2)=8+4f(-2) = \sqrt{8 + 4}f(−2)=8+4​ Adding 888 and 444 gives us 121212.

STEP 6

f(−2)=12f(-2) = \sqrt{12}f(−2)=12​ We can simplify 12\sqrt{12}12​ to 232\sqrt{3}23​.

STEP 7

f(−2)≈3.46f(-2) \approx 3.46f(−2)≈3.46 Rounding to two decimal places, we get **3.46**!

STEP 8

Now, let's **tackle** x=−1x = -1x=−1!Substituting −1-1−1 for xxx in f(x)f(x)f(x):

STEP 9

f(−1)=−4⋅(−1)+4f(-1) = \sqrt{-4 \cdot (-1) + 4}f(−1)=−4⋅(−1)+4​ −4-4−4 times −1-1−1 is 444.

STEP 10

f(−1)=4+4f(-1) = \sqrt{4 + 4}f(−1)=4+4​ 444 plus 444 is 888.

STEP 11

f(−1)=8f(-1) = \sqrt{8}f(−1)=8​ Simplifying 8\sqrt{8}8​ gives us 222\sqrt{2}22​.

STEP 12

f(−1)≈2.83f(-1) \approx 2.83f(−1)≈2.83 Rounding to two decimal places, we get **2.83**!

STEP 13

Time for x=0x = 0x=0!Substituting 000 for xxx in f(x)f(x)f(x):

STEP 14

f(0)=−4⋅0+4f(0) = \sqrt{-4 \cdot 0 + 4}f(0)=−4⋅0+4​ Anything multiplied by 000 is 000.

STEP 15

f(0)=0+4f(0) = \sqrt{0 + 4}f(0)=0+4​ 000 plus 444 is 444.

STEP 16

f(0)=4f(0) = \sqrt{4}f(0)=4​ The square root of 444 is **2**!

STEP 17

On to x=1x = 1x=1!Substituting 111 for xxx in f(x)f(x)f(x):

STEP 18

f(1)=−4⋅1+4f(1) = \sqrt{-4 \cdot 1 + 4}f(1)=−4⋅1+4​ −4-4−4 times 111 is −4-4−4.

STEP 19

f(1)=−4+4f(1) = \sqrt{-4 + 4}f(1)=−4+4​ −4-4−4 plus 444 is 000.

STEP 20

f(1)=0f(1) = \sqrt{0}f(1)=0​ The square root of 000 is **0**!

STEP 21

Lastly, let's **conquer** x=2x = 2x=2!Substituting 222 for xxx in f(x)f(x)f(x):

STEP 22

f(2)=−4⋅2+4f(2) = \sqrt{-4 \cdot 2 + 4}f(2)=−4⋅2+4​ −4-4−4 times 222 is −8-8−8.

STEP 23

f(2)=−8+4f(2) = \sqrt{-8 + 4}f(2)=−8+4​ −8-8−8 plus 444 is −4-4−4.

STEP 24

f(2)=−4f(2) = \sqrt{-4}f(2)=−4​ Uh oh!We can't take the square root of a negative number!

STEP 25

f(−2)≈3.46f(-2) \approx 3.46f(−2)≈3.46 f(−1)≈2.83f(-1) \approx 2.83f(−1)≈2.83 f(0)=2f(0) = 2f(0)=2f(1)=0f(1) = 0f(1)=0f(2)f(2)f(2) is undefined (DNE)

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1. Evaluate $f(-2)$
2. Evaluate $f(-1)$
3. Evaluate $f(0)$
4. Evaluate $f(1)$
5. Evaluate $f(2)$

Let's kick things off with $x = -2$ !
We're going to substitute $-2$ for $x$ in our function $f(x) = \sqrt{-4x + 4}$ .
Remember, whatever we do on one side of the equation, we do on the other.

$f(-2) = \sqrt{-4 \cdot (-2) + 4}$ Multiplying $-4$ by $-2$ gives us positive $8$ .

$f(-2) = \sqrt{8 + 4}$ Adding $8$ and $4$ gives us $12$ .

$f(-2) = \sqrt{12}$ We can simplify $\sqrt{12}$ to $2\sqrt{3}$ .

$f(-2) \approx 3.46$ Rounding to two decimal places, we get 3.46!

Now, let's tackle $x = -1$ !
Substituting $-1$ for $x$ in $f(x)$ :

$f(-1) = \sqrt{-4 \cdot (-1) + 4}$ $-4$ times $-1$ is $4$ .

$f(-1) = \sqrt{4 + 4}$ $4$ plus $4$ is $8$ .

$f(-1) = \sqrt{8}$ Simplifying $\sqrt{8}$ gives us $2\sqrt{2}$ .

$f(-1) \approx 2.83$ Rounding to two decimal places, we get 2.83!

Time for $x = 0$ !
Substituting $0$ for $x$ in $f(x)$ :

$f(0) = \sqrt{-4 \cdot 0 + 4}$ Anything multiplied by $0$ is $0$ .

$f(0) = \sqrt{0 + 4}$ $0$ plus $4$ is $4$ .

$f(0) = \sqrt{4}$ The square root of $4$ is 2!

On to $x = 1$ !
Substituting $1$ for $x$ in $f(x)$ :

$f(1) = \sqrt{-4 \cdot 1 + 4}$ $-4$ times $1$ is $-4$ .

$f(1) = \sqrt{-4 + 4}$ $-4$ plus $4$ is $0$ .

$f(1) = \sqrt{0}$ The square root of $0$ is 0!

Lastly, let's conquer $x = 2$ !
Substituting $2$ for $x$ in $f(x)$ :

$f(2) = \sqrt{-4 \cdot 2 + 4}$ $-4$ times $2$ is $-8$ .

$f(2) = \sqrt{-8 + 4}$ $-8$ plus $4$ is $-4$ .

$f(2) = \sqrt{-4}$ Uh oh!
We can't take the square root of a negative number!

$f(-2) \approx 3.46$ $f(-1) \approx 2.83$ $f(0) = 2$ $f(1) = 0$ $f(2)$ is undefined (DNE)