Question 822538
<ul><li>In function notation, like f(x), the expression between the parentheses, the "x" in f(x), represents the input to the function.</li><li>The whole expression, "f(x)", represents the output from f when the input is x.</li><li>When you are given a formula for a function, like f(x) = 3-x-x^2, the expression on the right shows us what the function does with its input as it figures out what the output will be. For this particular function
f(x) = 3-x-x^2
tells us that the function f will subtract the input, x, from a 3 and then subtract the square of the input. The result is the output to the function.</li></ul>So f(x+h) represents the output from function f when the input is x+h. Since we have the formula for f(x) we can figure out what the output will be:
{{{f(x) = 3-x-x^2}}}
So
{{{f(x+h) = 3-(x+h)-(x+h)^2}}}
Note the use of parentheses when substituting in the x+h for the x. This is critical. With them we will, for example, subtract the whole x+h from 3 as we should. Without them, 3 - x+h ..., we would subtract the x but not the h.<br>
Simplifying...
{{{f(x+h) = 3-(x+h)-(x^2+2xh+h^2)}}}
{{{f(x+h) = 3-x-h-x^2-2xh-h^2)}}}
There are no like terms so no more simplifying can be done.<br>
Repeating this for f(x-h):
{{{f(x-h) = 3-(x-h)-(x-h)^2}}}
{{{f(x-h) = 3-(x-h)-(x^2-2xh+h^2)}}}
{{{f(x-h) = 3-x+h-x^2+2xh-h^2)}}}<br>
Now that we have f(x+h) and f(x-h) we can deal with:
{{{(f(x+h)-f(x-h))/h}}}
Substituting in the expressions we got for f(x+h) and f(x-h):
{{{((3-x-h-x^2-2xh-h^2)-(3-x+h-x^2+2xh-h^2))/h}}}
Again with the parentheses!
Simplifying...
{{{(3-x-h-x^2-2xh-h^2-3+x-h+x^2-2xh+h^2))/h}}}
A lot of the numerator cancels out because they are opposites.
{{{(cross(3)-cross(x)-h -cross(x^2)-2xh-cross(h^2)-cross(3)+cross(x)-h+cross(x^2)-2xh+cross(h^2))/h}}}
{{{(-h-2xh-h-2xh)/h}}}
Adding like terms:
{{{(-2h-4xh)/h}}}
Factoring out h:
{{{(h(-2-4x))/h}}}
The factors of h cancel:
{{{(cross(h)(-2-4x))/cross(h)}}}
leaving:
{{{-2-4x}}}