Math  /  Algebra

QuestionThe table below shows the function f(x)=2x5f(x)=-2 x-5. Fill in the missing values. \begin{tabular}{|r|r|} \hline x\mathbf{x} & \multicolumn{1}{|c|}{y\mathbf{y}} \\ \hline\square & -1 \\ \hline-1 & \\ \hline & \\ \hline & \\ \hline 1 & \\ \hline 2 & \\ \hline & \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? We need to plug some xx values into the function f(x)=2x5f(x) = -2x - 5 and find the corresponding yy values, and vice-versa! Watch out! Make sure to apply the negative sign correctly in the function, especially when xx is also negative.
Don't mix up xx and yy values!

STEP 2

1. Find xx when yy is 1-1.
2. Find yy when xx is 1-1.
3. Find yy when xx is 11.
4. Find yy when xx is 22.

STEP 3

We know that y=f(x)y = f(x), so we can set 1-1 equal to our function: 1=2x5-1 = -2x - 5

STEP 4

To get xx by itself, let's add 55 to both sides of the equation.
Remember, we're adding 55 to *both* sides to keep the equation balanced! 1+5=2x5+5-1 + 5 = -2x - 5 + 5 4=2x4 = -2x

STEP 5

Now, we can divide both sides by 2-2 to find our **xx value**: 42=2x2\frac{4}{-2} = \frac{-2x}{-2} 2=x-2 = xSo, when yy is 1-1, xx is 2-2!

STEP 6

Let's substitute x=1x = -1 into our function f(x)f(x): f(1)=2(1)5f(-1) = -2 \cdot (-1) - 5

STEP 7

A negative times a negative is a positive, so 2-2 times 1-1 is 22: f(1)=25f(-1) = 2 - 5 f(1)=3f(-1) = -3So, when xx is 1-1, yy is 3-3!

STEP 8

Let's substitute x=1x = 1 into our function f(x)f(x): f(1)=215f(1) = -2 \cdot 1 - 5

STEP 9

f(1)=25f(1) = -2 - 5 f(1)=7f(1) = -7 So when xx is 11, yy is 7-7!

STEP 10

Let's substitute x=2x = 2 into our function f(x)f(x): f(2)=225f(2) = -2 \cdot 2 - 5

STEP 11

f(2)=45f(2) = -4 - 5 f(2)=9f(2) = -9 So when xx is 22, yy is 9-9!

STEP 12

Here's our completed table:
\begin{tabular}{|r|r|} \hline x\mathbf{x} & \multicolumn{1}{|c|}{y\mathbf{y}} \\ \hline 2-2 & -1 \\ \hline -1 & -3 \\ \hline 1 & -7 \\ \hline 2 & -9 \\ \hline \end{tabular}

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