Question

find an expression for a rational function f(x) that satisfies the conditions: a slant asymptote of y = 2x, vertical asymptote at x = 1, and contains the point (0,6)

Answer 1

@wio can you help?

Answer 2

I suppose I can try.

Answer 3

To get vertical asymptote you want something like \[
\frac 1{x-1}
\]

Answer 4

For the slant asymptote you want to add \(2x\) to it: \[
2x+\frac 1{x-1}
\]

Answer 5

Now that numerator can be what we want it to me... let me change: \[
2x+\frac {a}{x-1}
\]

Answer 6

We plug in the point to solve for \(a\). \[
2(0)+\frac{a}{(0)-1}=6
\]

Answer 7

Solving for \(a\) gives us: \[
\frac{a}{-1}=6\implies a=-6
\]

Answer 8

So now that we have our equation, we simplify: \[
f(x) = 2x-\frac{6}{x-1}=\frac{2x(x-1)-6}{x-1}=\frac{2x^2-2x-6}{x-1}
\]

Answer 9

@studygeek15 It's basically all about reverse engineering.

Answer 10

Thank you so much!