Teaching Notes 8.9: Finding the Composite of Two Functions
To find the composite of two functions, students must substitute the range of one function into
the domain of the other function. Many students find the notation of this process confusing.- Explain that a function is a rule that assigns to each member of the domain exactly one mem-
ber of the range. Note that the domain is the input and the range is the output. - Provide your students with this example:f(x)= 2 x. Explain that this function doubles the
input. Note that the domain isxand the range is 2x. Ask your students to findf(−1),f(3),
andf(10). (The answers are−2, 6, and 20.) - Explain that when the range of a function is substituted for the domain in another function,
a composite function results. For example, suppose that the range ofg(x)=x+3 was placed
in theffunction above. Because theffunction doubles the input, the output is 2(x+3) or
2 x+6. This can be expressed asf(g(x))= 2 x+6. Note that the range ofgis substituted in
f(x) and the function that results is the composite offandgand is denoted byf◦g. - Ask your students to explain the rule that applies tog(x)=x+3, which is that this function
adds 3 to the input. If the range off(x)= 2 xwas placed in thegfunction, 3 would be added
to 2x. This can be expressed asg(f(x))= 2 x+3. Note that the range offis substituted in
g(x) and the function that results is the composite ofgandfand is denoted byg◦f. - Review the information and example on the worksheet with your students. Emphasize that
the range of the function that is substituted is enclosed in the outermost parentheses.
EXTRA HELP:
Correct substitution is very important.f(g(x)) may not equalg(f(x)).ANSWER KEY:
(1) 3 x^2 + 1 (2) 16 x^2 − 16 x+ 4 (3) 12 x− 5 (4) 12 x+ 2 (5) 4 x^2 (6)− 6 x+ 1
(7) 9 x^2 + 6 x+ 1 (8)− 8 x− 2 (9)− 2 x^2 (10)− 6 x− 2 (11) 4 x^2 − 2 (12)− 8 x+ 4
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(Challenge)Bridget is correct.j(k(x))=k(j(x))=x.
------------------------------------------------------------------------------------------290 THE ALGEBRA TEACHER’S GUIDE