Question

The graph of the function f(x) = (x – 4)(x + 1) is shown below. Which statement about the function is true? The function is increasing for all real values of x where x < 0. The function is increasing for all real values of x where x < –1 and where x > 4. The function is decreasing for all real values of x where –1 < x < 4. The function is decreasing for all real values of x where x < 1.5.

Answer 1

we have

`f(x)=(x-4)(x+1)`

using a graph tool

see the attached figure

we know that

The function is increasing for all real values of x where
`x > 1.5`

and

The function is decreasing for all real values of x where
`x < 1.5`

therefore

Statements

case a) The function is increasing for all real values of x where x < 0

The statement is false

case b) The function is increasing for all real values of x where x < –1 and where x > 4

The statement is false

case c) The function is decreasing for all real values of x where –1 < x < 4

The statement is false

case d) The function is decreasing for all real values of x where x < 1.5

The statement is true

Answer 2

The correct option is:

The function is decreasing for all real values of x where x < 1.5.

Explanation:

We are given a function f(x) which is a product of two linear polynomial as:

`f(x)=(x-4)(x+1)`

Now, we know that on multiplying the two linear polynomial we will obtain a quadratic polynomial.

So, the function f(x) will be represented as:

`f(x)=x(x+1)-4(x+1)\\\\f(x)=x^2+x-4x-4\\\\f(x)=x^2-3x-4`

So, we will plot the graph of the function and check which statements about the function hold true.

1)

The function is increasing for all real values of x where x < 0.

This statement is false.

Since we get from the graph that function f(x) is decreasing for x<0.

2)

The function is increasing for all real values of x where x < –1 and where x > 4.

This option is incorrect as the function is decreasing for x<-1

whereas it is increasing for x>4.

3)

The function is decreasing for all real values of x where –1 < x < 4.

This option is incorrect.

Since, the function is both decreasing as well as increasing in the interval (-1,4).

4)

The function is decreasing for all real values of x where x < 1.5.

This option is correct.

Since it could be observed from the graph that the function is decreasing for x<1.5.