Question

For the function $y=3x+2,$ how do you find the set of coordinates with $x$-values $1$ and $3?$

Answer 1

Hint: For a function, the set of coordinates can easily be found when the $x$-values are given. What we have to do is to apply the values given in the given function. When we apply the $x$-values, we will get the corresponding $y$-values.

Complete step by step solution:
Consider the given function, $y=3x+2.$
We are asked to find the set of coordinates of the function when the $x$-values are given.
The given $x$-values are $1$ and $3.$
We have to find the corresponding $y$-values.
To find the $y$-values corresponding to the given $x$-values, we have to substitute the values of $x$-coordinates in the expression of the function.
Let us consider the $x$-value $1.$
That is $x=1.$
Let us find the $y$- value corresponding to \[x=1.\]
We are going to substitute $x=1$ in $y=3x+2$
Then we will get the $y$-value corresponding to $x=1$ as,
$\Rightarrow y=3\times 1+2$
When we multiply $3$ with $1,$ we will get \[3.\]
So, the above equation becomes,
$\Rightarrow y=3+2$
Here, when we add these numbers and get the sum, we will get the $y$-coordinate.
So, our $y$-coordinate corresponding to the $x$-value $x=1$ is,
 $\Rightarrow y=\left( 3+2 \right)=5.$
Therefore, the coordinate of the given function when the $x$-value equals to $1$ is $y=5.$
The coordinate point is $\left( 1,5 \right).$
Now, we consider the $x$-value $3.$
That is, $x=3.$
Let us find the $y$-value corresponding to the $x$-value $x=3.$
Now, we are substituting the $x$-value $x=3$ in the function $y=3x+2.$
Then, we will get the $y$-coordinate corresponding to $x=3$ as,
$\Rightarrow y=3\times 3+2.$
We know that $3\times 3=9.$
So, we will get
$\Rightarrow y=9+2=11$
Therefore, the coordinate of the given function when the $x$-value equals to $3$ is $y=11.$
The coordinate point is $\left( 3,11 \right).$

Hence, the set of coordinates for the function $y=3x+2$ when the \[x\]-values are $1$ and $3$ is $\left\{ \left( 1,5 \right),\left( 3,11 \right) \right\}.$

Note: We will be able to find all the coordinate points of a function when the $x$-coordinates are given. Similarly, when the $y$-coordinates are given, then with some rearrangements made in the expression of the function we can find the $x$-coordinates. We just have to convert the equation in terms of $y$ instead of $x.$