Define function based on asymptotes and intercept

Question

I'm looking for a function with the following characteristics:

• Vertical asymptote at $0$ (i.e. function never touches negative $x$-values)
• Horizontal asymptote at $7$ (i.e. function never results in $y$-values larger than $7$)
• $x$-intercept at $0.25$ (i.e. function crosses $x$-axis at $(0.25,0)$)

I tried working with a log-function (e.g. $y=\ln(x+0.75)$). This generally helped to achieve the intercept, but I still couldn't make the asymptotes work. Any advice?

Thanks

Answer 1

Define the function:

$$ y = \begin{cases} undefined & x\leq 0 \\ 7(1-\frac{1}{4x}) & x > 0 \end{cases} $$

This statisfies your requirements.

A logarithmic function is growing continuously and will never work. It has no upper bound.

A few more points:

• A horizontal asymptote does not mean that y values cannot be higher than the value of the asymptote.
• A vertical asymptote does not mean that x values cannot be lower than the value of the asymptote.

So, according to that, the function below also satisfies your requirements: $$y = 7(1-\frac{1}{4x})$$