Composition of Functions
Symbolic Composition
Composition of functions can also be evaluated symbolically. To evaluate a symbolic composition, first substitute a variable as the input of the innermost function and then use that output as the input of the next function. In this case, it is not the intent to find a certain numerical value. Instead, the composite function is the formula that results from substituting the formula for g(x) into the formula for f( x).
Example 1 Given f(x) = 2x + 3 and g(x) = -x2 + 5 , find (f o g)(x) .
(f o g)(x) = f(g(x))
= f(-x2 + 5)
Step 1. Rewrite the outer-function, leaving empty parentheses for the input expression.
f( ) = 2( ) + 3 ... setting up to insert the input formula
Step 2. Insert the function g(x) into the parentheses as the input expression.
f(-x 2 +5)= 2(-x2 + 5) + 3
Step 3. Simplify the right side of the equals sign.
f (-x2 + 5)= -2x2 + 10 + 3
f(g(x))= -2x2 + 13