Linear Functions: Problems with Solutions

Solution:
Answer: (D) None of these is correct.

Point A has coordinates $(5, 400)$. It means that for 5 hours we travel 400 miles.

Solution:
The $x$ axis shows hours and $y$ axis the pay.
The point $(1,40)$ shows that if he works one hour he will be paid \$40.
The point $(2,80)$ shows that if he works 2 hours he will be paid \$80.
To know the payments per hour, we must measure the vertical distance between the points $(1,40)$ and $(2,80)$.

Distance $=y_{2}-y_{1} = 80-40$

The payment is 40 dollars per hour.

Solution:
Correct: (B) A taxi service of 20 miles has a cost of \$8.

The point A $(20, 8)$ shows that the cost of the taxi service of 20 miles is 8 dollars.

Solution:
The definition of a function is the following:

For each point of the starting set corresponds exactly one value of the arrival set. In the table this condition is met, for every value of x there are no two different values of y, therefore this table represents a function that relates the variable x, y.

Solution:
Answer: (B) TABLE II

The definition of a function is the following:
to every element of the first set corresponds exactly one value of the second set.

In TABLE I the pairs [tex](-1,0)[/tex] and [tex](-1,1)[/tex] do not satisfy this condition, because for [tex]x=-1[/tex] there are two different values of [tex]y[/tex].
In TABLE III the pairs [tex](25,35)[/tex], [tex](25,24)[/tex] and [tex](25,28)[/tex] give three different values of [tex]y[/tex] for [tex]x=25[/tex].

In TABLE II the pairs that repeat are identical, [tex](-1,0)[/tex] twice and [tex](0,3)[/tex] twice, so every value of [tex]x[/tex] still has exactly one value of [tex]y[/tex]. TABLE II is a function.

Solution:
Both tables represent functions. In the first case we see the function identity to each value of x, the same value of y corresponds to it, this function is called identity function. In the second table we have the constant function, to all the values of x corresponds the same value of y, in both cases we have a function.

Solution:
For each element of $x$ corresponds a single element of $y$, for this reason the set of ordered pairs represents a function from set $A$ to set $B$

Solution:
In the linear function $y=2x-1$ we must substitute the value of $x$ by 1. $y=2(1)-1\Longrightarrow y=1$
Note that the graph of the function passes through the point $(1,1)$, but does not go through the point $(1,3)$. Then we can conclude that the point $(1,3)$ does not belongs to the graph of the function $y=2x-1$

Solution:
The correct answer is (B) $(1,3)$ and $(0,2)$.
$y=(1)+2 = 3 $ and
$y=(0)+2 =2$
So the points $(1,3)$ and $(0,2)$ belong to the graph of the function $y=x+2$.

You can check that none of the other points belong to the graph.

Solution:
We must replace x by 5 in the definition of the function $y=3x-\frac{2}{5}$

$y=3(5)-\frac{2}{5}=15-\frac{2}{5}=\frac{75-2}{5}=\frac{73}{5}$

Solution:
We must replace x by 1
$y=\frac{2(1)-1}{3}=\frac{1}{3}$

Solution:
Correct answer: (A) $(0,-2)$ and $(1,3)$

We must check if in the function $y=5x-2$ we replace x by 0 we get $y=-2$ and
if we replace x by 1 we get $y=3$.

For x = 0 we get
$y=5(0)-2 = 0-2 =-2$. So the point $(0,-2)$ satisfies the function $y=5x-2$

and also $y=5(1)-2=5-2=3$ which shows that the point $(1,3)$ satisfies the function.