Question

For the rational function f(x)=5−xx2+5x+6, find the points on the graph at the function value f(x)=3.

Answer 1

Given:

The rational function is:

$f(x)=\dfrac{5-x}{x^2+5x+6}$

To find:

The points on the graph at the function value $f(x)=3$.

Solution:

We have,

$f(x)=\dfrac{5-x}{x^2+5x+6}$

Substituting $f(x)=3$, we get

$3=\dfrac{5-x}{x^2+5x+6}$

$3(x^2+5x+6)=5-x$

$3x^2+15x+18=5-x$

Moving all the terms on one side, we get

$3x^2+15x+18-5+x=0$

$3x^2+16x+13=0$

Splitting the middle term, we get

$3x^2+3x+13x+13=0$

$3x(x+1)+13(x+1)=0$

$(3x+13)(x+1)=0$

Using zero product property, we get

$(3x+13)=0\text{ or }(x+1)=0$

$x=-\dfrac{13}{3}\text{ or }x=-1$

Therefore, the required values are $-\dfrac{13}{3},-1$.