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the function f(x) = -x^2 + 4x - 3 is graphed in the xy-coordinate plane as shown. based on the graph of the function which of the following statements are true? select all that apply. A. F(x) is less than 0 on the interval x is less than 0 B. F(x) is greater than 0 on the interval x is less than 0 C. F(x) is less than 0 on the interval 0 is less than x and x is less than 1 D. F(x) is greater than 0 on the interval 0 is less than x and x is less than 1 E. F(x) is less than 0 on the interval 1 is less than x and x is less than 3 F. F(x) is greater than 0 on the interval 1 is less than x and x is pess than 3 G. F(x) is less than 0 on the interval x is greater than 3 H. F(x) is greater than 0 on the interval x is greater than 3

Respuesta :

f(x) < 0 on the interval x < 0.
f(x) < 0 on the interval 0 < x < 1
f(x) > 0 on the interval 1 < x < 3
f(x) < 0 on the interval x > 3

Answer:

The correct options are:

  • A) F(x) is less than 0 on the interval x is less than 0.
  • C) F(x) is less than 0 on the interval 0 is less than x and x is less than 1.
  • F) F(x) is greater than 0 on the interval 1 is less than x and x is less than 3.
  • G) F(x) is less than 0 on the interval x is greater than 3.

Step-by-Step explanation:

We are given a function f(x) as:

[tex]F(x)=-x^2+4x-3[/tex]

This function f(x) could also be written as:

[tex]F(x)=-x^2+3x+x-3\\\\F(x)=-x(x-3)+1(x-3)\\\\F(x)=(1-x)(x-3)[/tex]

Now we will do each of the options:

A)

when x<0

then (1-x)>0 and (x-3)<0

Hence,

F(x)<0.

it could also be seen in the graph.

Hence, option A is correct.

B)

option: B is the opposite of option A and hence is false.

C)

In the interval 0<x<1.

we see that the graph of the function F(x) lie below the x-axis.

Hence, F(x) <0

Hence, option: C is correct.

D)

option: D is opposite of option C and hence is false.

E)

Clearly from the graph we could observe that:

F(x)>0 in the interval 1<x<3.

Hence, option E is incorrect.

F)

Option: F is opposite of option: E and hence is true.

G)

Again in the interval x>3 the graph of the function is below x-axis.

and hence F(x)<0 in the interval x>3.

Hence, option: G is correct.

H)

option: H is false.

since F(x)<0 in the interval x>3.

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