Math  /  Algebra

QuestionComplete the table. \begin{tabular}{|c|c|} \hline \multicolumn{2}{|c|}{f(x)=6x+3f(x)=-6|x|+3} \\ \hlinexx & f(x)f(x) \\ \hline-1 & \square \\ \hline 0 & \square \\ \hline 1 & \square \\ \hline 2 & \square \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? We need to plug some xx values into a function with absolute values and see what pops out! Watch out! Don't forget how absolute value works: it makes anything inside it positive (or zero if it's already zero)!

STEP 2

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

STEP 3

Let's **plug in** x=1x = -1 into our function f(x)=6x+3f(x) = -6|x| + 3.
So we have f(1)=61+3f(-1) = -6|-1| + 3.

STEP 4

Remember, the **absolute value** of a number is its distance from zero, so it's always positive (or zero)!
So, 1=1| -1 | = 1.

STEP 5

Now we have f(1)=61+3f(-1) = -6 \cdot 1 + 3.

STEP 6

Multiplying gives us 6+3-6 + 3.

STEP 7

Finally, adding gives us f(1)=3f(-1) = -3.
Awesome!

STEP 8

Now, let's **plug in** x=0x = 0.
We get f(0)=60+3f(0) = -6|0| + 3.

STEP 9

The absolute value of 00 is just 00, so we have f(0)=60+3f(0) = -6 \cdot 0 + 3.

STEP 10

Anything multiplied by zero is zero, so we're left with f(0)=0+3f(0) = 0 + 3, which simplifies to f(0)=3f(0) = 3.
Fantastic!

STEP 11

Let's **plug in** x=1x = 1.
We have f(1)=61+3f(1) = -6|1| + 3.

STEP 12

The absolute value of 11 is 11, so f(1)=61+3f(1) = -6 \cdot 1 + 3.
This looks familiar!

STEP 13

Multiplying gives us 6+3-6 + 3.

STEP 14

Adding those together, we find f(1)=3f(1) = -3.
Cool!

STEP 15

Time for our last value!
Let's **plug in** x=2x = 2.
We get f(2)=62+3f(2) = -6|2| + 3.

STEP 16

The absolute value of 22 is 22, so f(2)=62+3f(2) = -6 \cdot 2 + 3.

STEP 17

Multiplying gives us f(2)=12+3f(2) = -12 + 3.

STEP 18

Finally, adding gives us f(2)=9f(2) = -9.
Wonderful!

STEP 19

Our completed table looks like this:
\begin{array}{cc} \multicolumn{2}{c}{f(x)=-6|x|+3} \\ x & f(x) \\ -1 & -3 \\ 0 & 3 \\ 1 & -3 \\ 2 & -9 \\ \end{array}

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