Problem Evaluate the difference quotient for f (x) = x2 – 2x + 4.
This is a little different from but works the same way. Since a value is not supplied for x, we just leave it and work out the limit. Start by evaluating f (x + h):
Make sure you FOIL the square out and distribute the negative.
Now put this along with f (x) into the difference quotient.
As h gets smaller and smaller, the term in the middle gets smaller. This means the limit is equal to 2x – 2. Since the other terms do not contain x, they are unaffected when h gets small.
A terrific example of piecewise functions is our graduated income tax system. In that system, the more you make…the higher percentage you pay. However, you DO NOT pay the higher percentage on all of your income. In the two FAQ’s below,we take a look how all of this works.
Suppose you are asked to determine whether a function is discontinuous. Many of you might use technology to help you graph a function to decide what the limits are from the left and right. Remember, a function is continuous at a point if the limits from the left and right are equal and also match the value of the function at the point.
Be aware that the TI calculators, WolframAlpha, and Desmos may give slightly different graphs and lead you to the wrong conclusion.
The new MathFAQ below demonstrates how to graph
in WolframAlpha and Desmos.
Notice how the graphs differ. Which one is the better graph to use if you are deciding if the function is discontinuous?
Postage on first class mail in the United States is based on weight. Each ounce is charged according to a table published by the US Postal Service. The FAQ below shows how to take this table and write out a piecewise function P(x), where x is the weight of the letter.
I just saw yesterday that the IRS is starting to receive tax returns for the tax year 2016…YIPPEE! In honor of this auspicious moment, I thought I would add a FAQ about how piecewise functions and taxes are related.
for f (x) = x2 – 2x + 4.
but works the same way. Since a value is not supplied for x, we just leave it and work out the limit. Start by evaluating f (x + h):
A terrific example of piecewise functions is our graduated income tax system. In that system, the more you make…the higher percentage you pay. However, you DO NOT pay the higher percentage on all of your income. In the two FAQ’s below,we take a look how all of this works.
