X. F(x)
4. 13
2. 7
0. 1
-1. -2
using the table, what is the value of f(-1)?
Write a linear function equation for this table of values.
Using this equation or the table itself, find the value of f(1)
I don't understand this.
X. F(x)
4. 13
2. 7
0. 1
-1. -2
using the table, what is the value of f(-1)?
Write a linear function equation for this table of values.
Using this equation or the table itself, find the value of f(1)
I don't understand this.
K so the question is saying that there is a function $\displaystyle y=f(x)$ such that plugging in any x into the function will give you a value for y=f(x) as listed in the table.
Using this, we can think of the table as telling us this:
$\displaystyle f(4)=13 $
$\displaystyle f(2)=7$
$\displaystyle f(0) = 1 $
$\displaystyle f(-1) = -2$
Now as for making a function. What can we decide from the table? It looks like if x =0, f(x)=f(0)=1. So it looks like we need to have a constant that isn't multiplied by any x.
Next, lets looks at the values, how can we turn 2 into 7 and 4 into 13?
Looks like $\displaystyle f(x) = 3x + 1$. Does this make sense how we came to this? It can be a bit of a trial and error process.
Let's test it.
$\displaystyle f(2) = 3(2) + 1 = 6 + 1 = 7 \ \ \ \text{Check!}$
$\displaystyle f(4) = 3(4) + 1 = 12 +1 = 13 \ \ \ \text {Check!}$
$\displaystyle f(0) = 3(0) + 1 = 0 +1 = 1 \ \ \ \text {Check!}$
$\displaystyle f(-1) = 3(-1) + 1 = -3 +1 = -2 \ \ \ \text {Check!}$
Can you find $\displaystyle f(1)$ now?
Nice avatar, Kasper !
Otherwise, you might want to consider a more straightforward algebraïc way.
Your problem strongly suggests that a linear relationship exists between $\displaystyle x$ and $\displaystyle f(x)$. This is equivalent to saying that $\displaystyle f(x) = ax + b$, for some $\displaystyle a$ and $\displaystyle b$, $\displaystyle a \neq 0$.
1. Find $\displaystyle b$
You have some values of the function (actually, only two are required). You can say that :
$\displaystyle 13 = 4a + b$
$\displaystyle 7 = 2a + b$
We can slightly reformulate this :
$\displaystyle 13 - b = 4a$
$\displaystyle 7 - b = 2a$
And now, we get smart ! We divide both equations together :
$\displaystyle \frac{13 - b}{7 - b} = \frac{4a}{2a}$
Which is equivalent to :
$\displaystyle \frac{13 - b}{7 - b} = 2$
Now we simplify the fraction :
$\displaystyle 13 - b = 2(7 - b)$
$\displaystyle 13 - b = 14 - 2b$
$\displaystyle - b + 2b = 14 - 13$
$\displaystyle \boxed{b = 1}$
2. Find $\displaystyle a$
Now that we know $\displaystyle b$, this is easy : we know that $\displaystyle b = 1$ : we can use a pair of values $\displaystyle \left ( x, f(x) \right )$ :
$\displaystyle 13 = 4a + b$
Since $\displaystyle b = 1$, we get :
$\displaystyle 13 = 4a + 1$
$\displaystyle 12 = 4a$
$\displaystyle 3 = a$
$\displaystyle \boxed{a = 3}$
3. Conclusion
You are done : you have found the linear relationship between $\displaystyle x$ and $\displaystyle f(x)$ :
$\displaystyle \boxed{f(x) = 3x + 1}$
Let's check those results :
$\displaystyle f(4) = 4 \times 3 + 1 = 12 + 1 = 13$ -->
$\displaystyle f(2) = 2 \times 3 + 1 = 6 + 1 = 7$ -->
$\displaystyle f(0) = 0 \times 3 + 1 = 0 + 1 = 1$ -->
$\displaystyle f(-1) = (-1) \times 3 + 1 = -3 + 1 = -2$ -->
Now that we can find $\displaystyle f(x)$ from $\displaystyle x$ with a simple linear formula, let's answer the last part of the question : substitute $\displaystyle x = 1$ to find the value of $\displaystyle f(1)$ :
$\displaystyle f(1) = 1 \times 3 + 1 = 3 + 1 = 4$
Done !
_________________
Does it make sense ?
Hello, brianfisher1208!
. . . $\displaystyle \begin{array}{c|c}x & f(x) \\ \hline\text{-}1 & \text{-}2 \\ 0 & 1 \\ 2 & 7 \\ 4 & 13 \end{array}$
Using the table, what is the value of $\displaystyle f(-1)$ ?
Um . . . -2 ?
A linear function has the form: .$\displaystyle f(x) \;=\;ax+b$Write a linear function equation for this table of values.
And we must determine $\displaystyle a$ and $\displaystyle b.$
We can use any two values from our table.
For example:
. . $\displaystyle \begin{array}{ccccccccccccc}f(2) = 7: && a(2) + b &=& 7 && \Rightarrow && 2a + b &=& 7 & [1] \\
f(4) = 13: && a(4) + b &=& 13 && \Rightarrow && 4a + b &=& 13 & [2] \end{array}$
Subtract [2] - [1]: .$\displaystyle 2a \:=\:6 \quad\Rightarrow\quad a \:=\:3$
Substitute into [1]: .$\displaystyle 2(3) + b \:=\:7 \quad\Rightarrow\quad b \:=\:1$
Therefore: .$\displaystyle \boxed{f(x)\;=\;3x+1}$
Using this equation or the table itself, find the value of $\displaystyle f(1).$
$\displaystyle f(1) \;=\;3(1) + 1 \;=\;4$
Interesting! As you can see, Brian, Soroban chose to solve by substitution a system of linear equations in order to find $\displaystyle a$ and $\displaystyle b$, while I decided to arrange the equations to allow a division that will get rid of an unknown, making solving easy. Two equivalent solutions to one problem (although mine was a bit longer).
Thanks Soroban, I didn't know the command \boxed, I was sort of deceived my function didn't appear in math font