Write an equation of a rational function with these conditions: No Vertical Asymptote Horizontal Asymptote at y=5 Y-intercept at (0,3)

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The equation for rational function is 

`f(x) = [p(x)]/[g(x)]`

The function has no vertical asymptote. So, the denominator has no real roots. The simplest polynomial we can write is `x^2 + 1`

`=gt f(x) = [p(x)]/(x^2 +1)`

The function has horizontal asymptote at 5 and is greater than 0. So, both numerator and denominator have same degree and numerator is with coefficient equal to 5. 

`p(x) = 5x^2 + ax + b`

Here y- intercepts are given as, `a = 0 and b = 3`

Now `p(x) = 5x^2 + 0.x + 3`

`therefore ` the required equation for the rational function = `(5x^2 +3)/(x^2 + 1)`

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The function has no vertical Asymptote means the denominator does not  equate to 0 at any value of x. In other words the polynomial should not have real roots. A simple form of this type function of function would be `ax^2+1` .

The function has a horizontal Asymptote at y=5. So the polynomial of the numerator would have a type like `5x^2+bx+c` . 

So from these data we can say the function is;

`f(x) = (5x^2+bx+c)/(ax^2+1)`

It is given that at x = 0 then y = 3.

`3 = c/1`

`c = 3`


`f(x) = (5x^2+bx+3)/(ax^2+1)`

So a and b can be any rational value where `a!=0` .


A simple form of the answer would be at a = 1 and b = 0;


`f(x) =(5x^2+3)/(x^2+1)`


So the answer can be given as;

`f(x) =(5x^2+3)/(x^2+1)`




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