Question

If f(x) is a continuous function defined for all real numbers, f(-2) = 10, f(-7) = –5, and f(x) = 0 for one and only one value of x, then which of the following could be that x value?
a. -3
b. -8
c. 0
d. 2

Answer 1

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Answer 2

f(-2) = 10, f(-7) = -5
let's think of them as two given points and we are trying to calculate a continuous function such as a line.

f(-2) = 10 the point is (-2,10)
f(-7) =-5 the point is (-7,-5)

find the equation in slope intercept form and set y equal to 0. Solve for x and there's your answer.

Answer 3

The function is a line because in your question says "f(x) = 0 for one and only one value of x" that pretty much defines a linear function

Answer 4

you have a continuous function \(\large\color{slate}{ f(x) }\),

(will assume continuous over the interval \(\large\color{slate}{ (-\infty,\infty)}\) and for real numbers)

such that:
\(\large\color{slate}{ f(-2) = 10 }\)
\(\large\color{slate}{ f(-7) = ~-5 }\)

for what value of x can \(\large\color{slate}{ f(x) = 0 }\) ?

Answer 5

slope is 3

Answer 6

(It is definitely explicit, but) I didn't see a word \(\Large\color{slate}{ \rm linear }\) in there.

Answer 7

I think though, that they want a linear function though.

Answer 8

The slope is 3 that is correct.

Answer 9

idk wht to do from here

Answer 10

So with a slope 3, and use a point (-2,10) :

\(\large\color{slate}{ y-y_1=m(x-x_1)}\)

Answer 11

plug in your information, and re-write in a slope intercept form.

Answer 12

y=3x+16

Answer 13

\(\large\color{slate}{ y-10=-3(x-(-2))}\)
\(\large\color{slate}{ y-10=-3(x+2)}\)
\(\large\color{slate}{ y-10=-3x-6}\) ....

it is not 16, see the err ?

Answer 14

there isnt a negative in front of the 3 for the slope right?

Answer 15

@SolomonZelman

Answer 16

oh, true that

Answer 17

SO you are correct then, it is \(\large\color{slate}{ y=3x+16}\)

Answer 18

so now wht

Answer 19

\(\large\color{slate}{ f(x)=0}\) would be an x intercept.
Set y=0 equal zero, and solve for x.

\(\large\color{slate}{ 0=3x+16}\)

Answer 20

-5.333333=x

Answer 21

and repeating

Answer 22

@SolomonZelman

Answer 23

@SolomonZelman

Answer 24

yes, -16/3

Answer 25

I would cite the exact value.

Answer 26

now what

Answer 27

But this again, is assuming that your function is linear.

Answer 28

If f(x) is a continuous function defined for all real numbers, f(-2) = 10, f(-7) = -5, and f(x) = 0 for one and only one value of x, then which of the following could be that x value?
a. -3
b. -8
c. 0
d. 2

Answer 29

not an option, so it is not (necessarily) a linear function.

Answer 30

How do you mean, it's not one of the answer choice provided?

Answer 31

@SolomonZelman

Answer 32

I mean that -16/3 isn't an option.

Answer 33

yes ik tht... so there are more steps, correct?

Answer 34

ohh okay

Answer 35

if you have points,
(-2,10)
(-7,-5)

and only think about f(x) is given that it is continuous, but not necessarily is linear, then,

Answer 36

it can be (technically any point.
Continuous means that it is explicit (not more than one x value for a y value)
and that it can be drawing without picking the pencil of the paper.