Unit
Introduction to Functions
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
There are actually a couple different kinds of difference quotients. Here is another kind that we haven’t seen yet. F(x) minus f(3) over x minus 3 and you can replace the three by any number it’s still a difference quotient. Let’s calculate that difference quotient for this function; f(x) equals the square root of x plus 1.
F(x) minus f(3) over x minus 3. F(x) is just root x plus 1, f(3) is root 3 plus 1, which is root 4, which is 2. So minus 2, over x minus 3.
When you have a difference quotient that involves a radical function you’re allowed to use the trick of multiplying by the conjugate. The conjugate of this would be root x plus 1 plus 2. You multiply the top and bottom by that. Because remember, when you’re manipulating an expression, you can't just multiply the top by a number. You’ll the change the value of the expression. You’re effectively multiplying by 1, when you multiply the top and the bottom by the same thing.
This becomes a difference of squares. You get root x plus 1 quantity squared, which is x plus 1 minus 2 squared, so minus 4, over x minus 3, times root x plus 1 plus 2. In the numerator, this becomes x minus 3 over x minus 3 times the quantity, root x plus 1, plus 2. And the x minus 3s cancel. You’re left with 1 over root x plus 1 plus 2 and that’s your answer.
When you’re dealing with a difference quotient that involves a radical function, remember this trick of multiplying the top and bottom by the conjugate of the numerator. And this is actually a very desirable form for when you are in calculus and you’re evaluating derivatives using a difference quotient.