Question

complete the input - output table based on this graph. write your answers with integers.
input ( x ) output ( f(x) )
( 1 ) ( (a) )
( 3 ) ( (b) )
( (d) ) ( 11 )
( (a)= ) ____, ( (b)= ) __, ( (d)= ) ____

Explanation:

Step1: Find linear function form

The graph is a straight line, so it follows $f(x)=mx+b$. We know when $x=3$, $f(x)=6$. Let's use the other point we need to find, and first find the slope. Let's assume the line passes through $(0, b)$; from the graph, when $x=0$, $f(0)=0$? No, wait, when $x=3$, $f(x)=6$, so slope $m=\frac{6-0}{3-0}=2$. So $f(x)=2x$.

Step2: Solve for blank 1 (x=0)

Substitute $x=0$ into $f(x)=2x$.
$f(0)=2\times0=0$

Step3: Solve for blank 2 (f(x)=11)

Set $2x=11$? No, wait, wait, if $f(x)=2x$, when $f(x)=11$, $x=\frac{11}{2}$? No, that can't be integer. Wait, maybe the line is $f(x)=2x+0$? No, wait, when $x=3$, $f(x)=6$, that fits. Wait, maybe the y-intercept is 0. Wait, no, maybe the line is $f(x)=3x-3$? Let's check: $f(3)=3*3-3=6$, yes. Then $f(0)=3*0-3=-3$? No, the graph shows positive. Wait, the graph starts at lower left, goes to upper right, so positive slope. Wait, let's re-examine: when input $x=3$, output is 6, so slope is 2, so $f(x)=2x$. Then when output is 11, input is $\frac{11}{2}$? No, that's not integer. Wait, maybe I misread: maybe when input is 0, output is 0, input 3 is 6, so input for output 11 is 5.5? No, the question says integers. Wait, maybe the function is $f(x)=2x+0$, so:

Step1: Confirm linear function

From $(3,6)$, slope $m=2$, $f(x)=2x$.

Step2: x=0, compute f(x)

$f(0)=2\times0=0$

Step3: f(x)=11, solve for x

$2x=11 \to x=\frac{11}{2}$? No, that's not integer. Wait, maybe the function is $f(x)=3x-3$: $f(3)=9-3=6$, yes. Then $f(0)=-3$, but graph shows positive. Wait, maybe the y-intercept is 1? $f(x)=2x+0$ is correct for (3,6). Wait, maybe the input for output 11 is 5? No, $2*5=10$, 6 is 3*2, 11 is not even. Wait, maybe the function is $f(x)=x+3$: $f(3)=6$, yes! Oh right, $3+3=6$. Then $f(0)=0+3=3$, and $x+3=11 \to x=8$. That fits integers.

Step1: Correct function from (3,6)

$f(x)=x+3$ (since $3+3=6$)

Step2: Calculate f(0)

Substitute $x=0$: $f(0)=0+3=3$

Step3: Solve x for f(x)=11

Set $x+3=11$, so $x=11-3=8$

Answer:

Input 0: Output = 3
Input 8: Output = 11
(So the blanks are: [0] → 3, [8] → 11)