Function Notation Math Example 3

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Example 3

medium
If f(x)=2xโˆ’1f(x) = 2x - 1 and g(x)=x2+3g(x) = x^2 + 3, find (fโˆ˜g)(x)(f \circ g)(x) and (gโˆ˜f)(x)(g \circ f)(x).

Solution

  1. 1
    (fโˆ˜g)(x)=f(g(x))=f(x2+3)=2(x2+3)โˆ’1=2x2+5(f \circ g)(x) = f(g(x)) = f(x^2+3) = 2(x^2+3) - 1 = 2x^2 + 5.
  2. 2
    (gโˆ˜f)(x)=g(f(x))=g(2xโˆ’1)=(2xโˆ’1)2+3=4x2โˆ’4x+1+3=4x2โˆ’4x+4(g \circ f)(x) = g(f(x)) = g(2x-1) = (2x-1)^2 + 3 = 4x^2 - 4x + 1 + 3 = 4x^2 - 4x + 4.

Answer

(fโˆ˜g)(x)=2x2+5,(gโˆ˜f)(x)=4x2โˆ’4x+4(f \circ g)(x) = 2x^2 + 5, \quad (g \circ f)(x) = 4x^2 - 4x + 4
Composition fโˆ˜gf \circ g means 'apply gg first, then ff.' The notation (fโˆ˜g)(x)=f(g(x))(f \circ g)(x) = f(g(x)) is read inside-out. Note that fโˆ˜gโ‰ gโˆ˜ff \circ g \neq g \circ f in general โ€” function composition is not commutative.

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.

Learn more about Function Notation โ†’

More Function Notation Examples

Example 1 easy

If [formula], find [formula].

Example 2 medium

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

Example 4 hard

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

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