A hole has to be 2 ft deep. William is able to dig at a rate of 1/2 ft. per minute. How do you write an equation for when y, the amount he has left to dig is a function of x, the amount of time he has been digging?

Sep 22, 2016

$f \left(x\right) = 2 - \frac{1}{2} x$

Explanation:

Since my name is William so I feel inclined to answer this question as it pertains to my digging capabilities.

In general any function can be expressed as $f \left(x\right) = y$. Now lets call the amount of time digging $x$ and that is measured in minutes. If this is the case we know that I can dig about about $\frac{1}{2}$ft. per minute so after 2 minutes the hole is 1 foot deep. If the end result is to get to 2ft then we just need to subtract that from what has already been done to get the amount of time left.

We can write the equation as $f \left(x\right) = 2 - \frac{1}{2} x$. Now lets check that this works. First we already noticed that after 2 minutes we only need to have 1 foot left so

$f \left(2\right) = 2 - \frac{1}{2} 2 = 1$, this is true

Next lets check if we work the whole 4 minutes if the amount left is 0
$f \left(4\right) 2 - \frac{1}{2} 4 = 0$. So this is true.

we could continue to check more if you like.