Question

If f(x) = (1/3)x + 9, which statement is always true?

(1) f(x) < 0 (3) If x < 0, then f(x) < 0.
(2) f(x) > 0 (4) If x > 0, then f(x) > 0.

Answer 1

`f(x)=(1)/(3)x+9 \\ \\ f(x)\ \textless \ 0 \\ (1)/(3)x+9\ \textless \ 0 \\ (1)/(3)x\ \textless \ -9 \\ x\ \textless \ -27 \\ \\ f(x)\ \textgreater \ 0 \\ (1)/(3)x+9\ \textgreater \ 0 \\ (1)/(3)x\ \textgreater \ -9 \\ x\ \textgreater \ -27`

f(x)<0 for x<-27, f(x)>0 for x>-27.
Therefore, if x<0 then f(x) can be either less or greater than 0. If x>0 then f(x) is greater than 0.

Statement (4) is always true.

Answer 2

`1) f(x) \ \textless \ 0`

=> We do not know the value of x. Therefore, we cannot be sure. Not sure.

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`2) f(x) \ \textgreater \ 0`

=> The same way. X is unknown, it is not always true.

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3) If x < 0, then f(x) < 0

=> It doesn't provide in each case.

Let's take two examples:

◘ Suppose that; x = -30


`(1)/(3) x + 9 = (1)/(3) (-30) + 9`


`= (-10) + 9 = -1`

Okay, it provided.

◘ But, x= -3


`(1)/(3) x + 9 = (1)/(3) (-3) + 9 = 8`

x <0 but f(x) > 0

False.

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4) x > 0, then f(x) > 0

That's true !

Examples:

x= 6


`(1)/(3) x + 9 = (1)/(3). 6 + 9 = 2 + 9 = 11`

f(6) = 11

Namely,

if x>0 , f(x) >0

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Answer= 4