Question

find the formula for a rational function that satisfies all of the following. its x-intercept is (-3,0). it has a vertical asymptotes, namely x=-1 and x=0. it has one horizontal asymptote, namely y=3

Answer 1

Let that be

`\\ \rm\Rightarrow y=(a(f(x)))/(g(x))`

Two vertical asymptotes at -1 and 0

`\\ \rm\Rightarrow y=(a(f(x))/(x(x+1))`

If we simply

`\\ \rm\Rightarrow y=(a(f(x))/(x^2+1)`

• Denominator has degree 2
• Numerator should have degree as 2 and coefficient as 3 inorder to get horizontal asymptote y=3 means the quadratic equation should contain 3x²
• But there should be a x intercept at -3 so one zeros should be -3

Find a equation

• 3x²+9x

Find zeros

• 3x²+9x=0
• 3x(x+3)=0
• x=0,-3

Horizontal asymptote

• 3x²/x²
• 3

So our equation is

`\\ \rm\Rightarrow y=(3x^2+9x)/(x(x+1))`

Graph attached