A relation in which each $x -$ value corresponds with exactly one $y -$ value is a function. To explain how we know that the patterns are functions, we will refer to each of the tables. In the tables, we will examine the input values and if each of them only corresponds to one output value, then we can conclude the relation is a function.

Table A

We are given that the rectangle has a width of

x

and a length of

2 x .

To find the perimeter of the rectangles, we need to add each of the sides.

P = 2 ℓ + 2 w

▼

Evaluate

SubstituteII

$ℓ = 2 x$ , $w = x$

P = 2 (2 x) + 2 (x)

Multiply

P = 4 x + 2 x

AddTerms

Add terms

P = 6 x

Now, we can substitute each of the given values to find the perimeter of the similar rectangles.

$x$	$1$	$2$	$3$	$4$	$5$
$6 x$	$6 (1)$	$6 (2)$	$6 (3)$	$6 (4)$	$6 (5)$
$P$	$6$	$12$	$18$	$24$	$30$

We can see that $P$ is different for every value of $x .$

Table B

Similarly to what we did for Table A, to find the areas of the rectangles, we will substitute the width and length in terms of

x

in the formula for the area of a rectangle.

A = ℓ w

SubstituteII

$ℓ = 2 x$ , $w = x$

A = (2 x) (x)

Multiply

A = 2 x^{2}

Now, we can substitute the values of

x

to complete the table!

$x$	$1$	$2$	$3$	$4$	$5$
$2 x^{2}$	$2 (1)^{2}$	$2 (2)^{2}$	$2 (3)^{2}$	$2 (4)^{2}$	$2 (5)^{2}$
$A$	$2$	$8$	$18$	$32$	$50$

Here, we can notice that the area of a rectangle is different for different values of $x .$

Table C

We are given that the radius of every circle is equal to

x .

To find an expression for the circumference, we will substitute

x

in the formula.

C = 2 π r \Rightarrow C = 2 π x

Now, we can substitute the values of

x

to fill the table.

$x$	$1$	$2$	$3$	$4$	$5$
$2 π x$	$2 π (1)$	$2 π (2)$	$2 π (3)$	$2 π (4)$	$2 π (5)$
$C$	$2 π$	$4 π$	$6 π$	$8 π$	$10 π$

We can see that $C$ is different for every value of $x .$

Table D

Now, we will substitute

x

for

r

in the formula for the area of a circle.

A = π r^{2} \Rightarrow A = π x^{2}

Let's substitute the values of

x

to fill the table.

$x$	$1$	$2$	$3$	$4$	$5$
$π x^{2}$	$π (1)^{2}$	$π (2)^{2}$	$π (3)^{2}$	$π (4)^{2}$	$π (5)^{2}$
$A$	$π$	$4 π$	$9 π$	$16 π$	$25 π$

Again, we can see that for every value of $x,$ we obtain a different value of $A .$

Conclusion

In each of the tables above, we can see that each input corresponds with exactly one output. Therefore, every pattern represents a function.

Exercise