Function Notation Math Example 2

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Example 2

medium
If g(x)=x2+3xg(x) = x^2 + 3x, find and simplify g(x+h)−g(x)h\frac{g(x+h) - g(x)}{h}.

Solution

  1. 1
    Find g(x+h)g(x+h): (x+h)2+3(x+h)=x2+2xh+h2+3x+3h(x+h)^2 + 3(x+h) = x^2 + 2xh + h^2 + 3x + 3h.
  2. 2
    g(x+h)−g(x)=(x2+2xh+h2+3x+3h)−(x2+3x)=2xh+h2+3hg(x+h) - g(x) = (x^2 + 2xh + h^2 + 3x + 3h) - (x^2 + 3x) = 2xh + h^2 + 3h.
  3. 3
    g(x+h)−g(x)h=h(2x+h+3)h=2x+h+3\frac{g(x+h) - g(x)}{h} = \frac{h(2x + h + 3)}{h} = 2x + h + 3.

Answer

2x+h+32x + h + 3
The difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h} is the foundation of derivatives in calculus. It represents the average rate of change of ff over the interval [x,x+h][x, x+h]. As h→0h \to 0, this approaches the instantaneous rate of change f′(x)f'(x).

About Function Notation

Function notation f(x)f(x) is a shorthand that names a function (ff) and specifies its input (xx). Writing f(3)=10f(3) = 10 means that when the input is 3, the function produces the output 10. This notation is not multiplication.

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More Function Notation Examples

Example 1 easy

If [formula], find [formula].

Example 3 medium

If [formula] and [formula], find [formula] and [formula].

Example 4 hard

A function satisfies [formula] for all [formula], with [formula]. Find [formula], [formula], and [fo

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