Name:
Date:
Algebra 2
Composites of Functions Using Formulas
Example
first function: f(x) = 2x + 1
second function: g(x) = x2
Find f(g(x)):
g(x) = x
f (x) = 2x +1
2
f (g(x)) = 2(x 2 )+1
f (x 2 ) = 2(x 2 )+1
Now find g(f(x)):
gx  x 2
f x   2x  1
g f x 2x 1
2
g2x  1  2x  1
2
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Try on Your Own
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Find these composite formulas.
1. Suppose f(x) = 5x – 3 and g(x) = 2x + 7.
a. Write the formula for g(f(x))
b. Write the formula for f (g(x))
2. Suppose f(x) = x3 and g(x) =
x.
a. Write the formula for f (g(x))
b. Write the formula for g( f (x))
Name:
Date:
Algebra 2
Composite Functions (Formulas) Homework Problems
1. For each pair of functions f(x) and g(x), write the formula for the composite g(f(x)).
a. f(x) = 4x – 7, g(x) = x2
b. f(x) = x3,
c. f(x) =
x,
g(x) = 2x + 6
g(x) = 1 + x
d. f(x) = 3x,
g(x) = 5x – 1
e. f(x) = 4x,
g(x) = x + x2
Hint: Replace every x in the g(x) formula with _______.
2. Consider the composite where the first function is f(x) = 3 – x and the second
is g(x) = 2x.
a. Evaluate g(f(1)). That is: Start with 1 as input. First use function f, then use
function g.
b. Evaluate g(f(3)).
c. Complete this input-output table for the composite. You found two of the
numbers already in parts a and b.
x
g(f(x))
1
2
3
4
5
d. Write the function formula for the composite: g(f(x)) =
Name:
Date:
Algebra 2
3. Use these functions for all problems on this page:
f(x) = x – 2, g(x) = x .
a. Evaluate g(f(6)).
b. Evaluate g(f(11)).
c. Fill in this input-output table for the composite. You will need to find some of the
needed square-roots on your calculator (OK to round to three decimal places).
x
g(f((x))
2
4
6
8
10
d. Try to evaluate g(f(1)). Something goes wrong. Explain.
e. The point of part d is that x = 1 cannot be in the domain for this composite
function.
What x values can be included in the domain?
f. Write the function formula for the composite: g(f(x)) =
Name:
Date:
Algebra 2
4. Use these functions for all problems on this page:
f(x) = 3x + 1,
g(x) = 10 – x.
Some of the questions below are about a composite where f is first and g is second,
while others are about a composite where g is first and f is second. Remember that the
function that’s on the right, closest to the x, comes first. So:
g(f(x)) means that f is first, g is second (“f followed by g”);
f(g(x)) means that g is first, f is second (“g followed by f”).
a. Evaluate g(f(2)). That is, begin with 2 as an input to f, and find the composite’s
output.
b. Evaluate f(g(2)). That is, begin with 2 as an input to g, and find the composite’s
output.
c. Evaluate g(f(5)).
d. Evaluate f(g(5)).
e. Explain why the function formula for the composite g(f(x)) is g(f(x)) = 10 – (3x +
1).
f. What is the function formula for f(g(x))?
g. Find the zero of g(f(x)). Hint: Solve 10 – (3x + 1) = 0.
h. Find the zero of f(g(x)).
Name:
Date:
Algebra 2
5. For each pair of functions f(x) and g(x), write formulas for both composites, g(f(x)) and
f(g(x)).
a. f(x) = 2x – 1, g(x) = 3x
g(f(x)) =
b. f(x) = x2,
f(g(x)) =
g(x) = x + 4
g(f(x)) =
c. f(x) = x – 5,
f(g(x)) =
g(x) =
2
x
g(f(x)) =
f(g(x)) =
6. Let f(x) = x – 5 and g(x) =
1
2
x.
a. Find g(f(3)), g(f(5)), and g(f(9)) by using function f followed by function g.
x
g(f((x))
show work here
3
f
g
3¾
¾®
____ ¾
¾®
____
5
f
g
5¾
¾®
____ ¾
¾®
____
9
f
g
9¾
¾®
____ ¾
¾®
____
b. Write the function formula for the composite:
g(f(x)) =
c. Find g(f(3)), g(f(5)), and g(f(9)) again, by evaluating the formula from part b.
x
g(f((x))
3
5
9
Answers to a and c should agree.
show work here
Name:
Date:
Algebra 2
7. Let f(x) = x2 and g(x) = 3 – x.
a. Complete this table by using function f followed by function g.
x
g(f((x))
show work here
f
g
-4¾
¾®
____ ¾
¾®
____
–4
–3
–2
–1
0
1
2
3
4
b. Write the function formula for the composite: g(f(x)) =
c. Complete this table again, by evaluating the formula from part b.
x
g(f((x))
–4
–3
–2
–1
0
1
2
3
4
Answers to a and c should agree.
show work here
Name:
Date:
Algebra 2