Given f of x is equal to the quantity x minus 4 end quantity divided by the quantity x squared plus 13x plus 36 end quantity, which of the following is true?

f(x) is decreasing for all x > –9
f(x) is increasing for all x < –9
f(x) is increasing for all x > –4
f(x) is decreasing for all x > –4

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shubhamchouhanvrVT shubhamchouhanvrVT · Answer 1 · 2022-06-28T13:22:32+02:00

The simplified expression of the function [tex]f(x) =\dfrac{x-4}{x^2+13x+36}[/tex] is

[tex]f(x) = \dfrac{x-4}{(x+9)(x+4)}[/tex]. f(x) is increasing for all x > –4.The correct answer is option C.

A function is defined as a relation between the set of inputs having exactly one output each.

The function expression is given as:

[tex]f(x) =\dfrac{x-4}{x^2+13x+36}[/tex]

Expand the denominator

[tex]f(x) =\dfrac{x-4}{x^2+4x +9x+36}[/tex]

Factorize the denominator

[tex]f(x) =\dfrac{x-4}{x(x+4)+9(x+4)}[/tex]

Factor out x + 9

[tex]f(x) = \dfrac{x-4}{(x+9)(x+4)}[/tex]

The simplified expression of the function [tex]f(x) =\dfrac{x-4}{x^2+13x+36}[/tex] is

[tex]f(x) = \dfrac{x-4}{(x+9)(x+4)}[/tex]. f(x) is increasing for all x > –4