SAT Question 12983c1e: Answer, Explanation & Desmos Solution

$x$	$f (x)$
$1$	$5$
$3$	$13$
$5$	$21$

Some values of the linear function $f$ are shown in the table above. Which of the following defines $f$ ?

Difficulty:Easy

$f (x) = 2 x + 3$

$f (x) = 3 x + 2$

$f (x) = 4 x + 1$

$f (x) = 5 x$

Explanation

Correct answer: C.

Choice C is correct. Because $f$ is a linear function of $x$ , the equation $f (x) = m x + b$ , where $m$ and $b$ are constants, can be used to define the relationship between $x$ and $f (x)$ . In this equation, $m$ represents the increase in the value of $f (x)$ for every increase in the value of $x$ by $1$ . From the table, it can be determined that the value of $f (x)$ increases by $8$ for every increase in the value of $x$ by $2$ . In other words, for the function $f$ the value of $m$ is $\frac{8}{2}$ , or $4$ . The value of $b$ can be found by substituting the values of $x$ and $f (x)$ from any row of the table and the value of $m$ into the equation $f (x) = m x + b$ and solving for $b$ . For example, using $x = 1$ , $f (x) = 5$ , and $m = 4$ yields $5 = 4 \times 1 + b$ . Solving for $b$ yields $b = 1$ . Therefore, the equation defining the function $f$ can be written in the form $f (x) = 4 x + 1$ .

Desmos solution:

Enter in Desmos:

$x_{1} = [1, 3, 5]$
$f_{1} = [5, 13, 21]$
$f_{1} \sim a x_{1} + b$
$f (x) = 4 x + 1$

Reviewed and rewritten by CookSAT

Why the other choices are wrong

Choice A

Choice A is incorrect. Any equation defining the linear function $f$ must give values of $f (x)$ for corresponding values of $x$ , as shown in each row of the table. According to the table, if $x = 3$ , $f (x) = 13$ . However, substituting $x = 3$ into the equation given in choice A gives $f (3) = 2 \times 3 + 3$ , or $f (3) = 9$ , not $13$ . Therefore, the equation in choice A cannot define $f$ .

Choice B

Choice B is incorrect. Any equation defining the linear function $f$ must give values of $f (x)$ for corresponding values of $x$ , as shown in each row of the table. According to the table, if $x = 3$ , $f (x) = 13$ . However, substituting $x = 3$ into the equation given in choice B gives $f (3) = 3 \times 3 + 2$ , or $f (3) = 11$ , not $13$ . Therefore, the equation in choice B cannot define $f$ .

Choice D

Choice D is incorrect. Any equation defining the linear function $f$ must give values of $f (x)$ for corresponding values of $x$ , as shown in each row of the table. According to the table, if $x = 3$ , $f (x) = 13$ . However, substituting $x = 3$ into the equation given in choice D gives $f (3) = 5 \times 3$ , or $f (3) = 15$ , not $13$ . Therefore, the equation in choice D cannot define $f$ .

SAT Question 12983c1e: Linear Functions answer and explanation with Desmos solution

Explanation

Enter in Desmos:

Why the other choices are wrong