Function Notation Math Example 4

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Example 4

hard
A function satisfies f(x+1)=2f(x)+3f(x+1) = 2f(x) + 3 for all xx, with f(0)=1f(0) = 1. Find f(1)f(1), f(2)f(2), and f(3)f(3).

Solution

  1. 1
    f(1)=2f(0)+3=2(1)+3=5f(1) = 2f(0) + 3 = 2(1) + 3 = 5. f(2)=2f(1)+3=2(5)+3=13f(2) = 2f(1) + 3 = 2(5) + 3 = 13. f(3)=2f(2)+3=2(13)+3=29f(3) = 2f(2) + 3 = 2(13) + 3 = 29.
  2. 2
    Pattern: f(n)=2n+2−3f(n) = 2^{n+2} - 3. Check: f(0)=4−3=1f(0) = 4-3=1 ✓, f(1)=8−3=5f(1) = 8-3=5 ✓, f(2)=16−3=13f(2) = 16-3=13 ✓, f(3)=32−3=29f(3) = 32-3=29 ✓.

Answer

f(1)=5,f(2)=13,f(3)=29f(1) = 5, \quad f(2) = 13, \quad f(3) = 29
Recursive function definitions use function notation to define values in terms of previous values. The relation f(x+1)=2f(x)+3f(x+1) = 2f(x) + 3 is a first-order linear recurrence. The closed form f(n)=2n+2−3f(n) = 2^{n+2} - 3 can be found by solving the recurrence or by observing the pattern.

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 2 medium

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

Example 3 medium

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

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