Evaluate The average rate of change of 𝑓 of 𝑥 equals negative one over eight 𝑥 minus five when 𝑥 varies from 𝑥 sub one to 𝑥 sub one plus ℎ.
The average rate of change formula allows us to find the rate at which the function’s output changes as compared to the function’s input. For our function 𝑦 equals 𝑓 of 𝑥 for values of 𝑥 from 𝑎 to 𝑏, the average rate of change is given by 𝑓 of 𝑏 minus 𝑓 of 𝑎 over 𝑏 minus 𝑎. Now we’re told that our function 𝑓 of 𝑥 is negative one over eight 𝑥 minus five. But what do we define 𝑎 and 𝑏 to be? Well, our input, our value of 𝑥, varies from 𝑥 sub one to 𝑥 sub one plus ℎ. So we’re going to let 𝑎 be equal to 𝑥 sub one and 𝑏 be equal to 𝑥 sub one plus ℎ.
Then we evaluate 𝑓 of 𝑎 and 𝑓 of 𝑏. We see 𝑓 of 𝑎 is 𝑓 of 𝑥 sub one. So we substitute 𝑥 for 𝑥 sub one. Similarly, 𝑓 of 𝑥 sub one plus ℎ is found by substituting 𝑥 sub one plus ℎ for 𝑥. And we get negative one over eight times 𝑥 sub one plus ℎ minus five. And if we distribute the parentheses, our denominator becomes eight 𝑥 sub one plus eight ℎ minus five.
Let’s substitute everything we know then into the average rate of change formula. When we do, we subtract our 𝑓 of 𝑥 sub one from 𝑓 of 𝑥 sub one plus ℎ. And so our numerator is as shown. Our denominator is 𝑏 minus 𝑎. So it’s 𝑥 sub one plus ℎ minus 𝑥 sub one. And so we notice that on the denominator 𝑥 sub one minus 𝑥 sub one is zero. And we’re simply left with ℎ. We’re going to need to do something with the numerator, though.
And so before we do, we notice that we’re subtracting a negative. So this is the same as adding the fractions. And to add algebraic fractions, we need to find a common denominator. The common denominator here will simply be the product of the denominators we have. We multiply the numerator and denominator of our first fraction by eight 𝑥 sub one minus five and of our second by eight 𝑥 one plus eight ℎ minus five. And that leaves us with the common denominator shown.
Since we’re subtracting eight 𝑥 sub one minus five, we get negative eight 𝑥 sub one plus five. And then we see that negative eight 𝑥 sub one plus eight 𝑥 sub one is zero and plus five minus five is zero. That leaves us with simply eight ℎ on our numerator. And so we can divide our ℎ’s. And so our average rate of change function simplifies really nicely. Eight ℎ divided by ℎ is eight. And so we get eight over eight 𝑥 sub one minus five times eight 𝑥 sub one plus eight ℎ minus five.