8.3 Composite Functions
Chapter
Chapter 8
Section
8.3
Lectures 15 Videos 5.52mins
Introduction to Composition of Functions

Below is an example of Even Function. Algebraically for y = f(x) needs to satisfy

 \displaystyle f(-x) = -f(x)  to be an even function.

4.49mins
Even Function
Solutions 18 Videos

Let f(x) = -x + 2 and g(x) = (x + 3)^2. Determine a simplified algebraic model for the composite function.

y = f(g(x))

0.36mins
Q1a

Let f(x) = -x + 2 and g(x) = (x + 3)^2. Determine a simplified algebraic model for the composite function.

y = g(f(x))

0.31mins
Q1b

Let f(x) = -x + 2 and g(x) = (x + 3)^2. Determine a simplified algebraic model for the composite function.

y = f(f(x))

0.30mins
Q1c

Let f(x) = -x + 2 and g(x) = (x + 3)^2. Determine a simplified algebraic model for the composite function.

y = g(g(x))

0.24mins
Q1d

Let f(x) = -x + 2 and g(x) = (x + 3)^2. Determine a simplified algebraic model for the composite function.

y = f^{-1}(f(x))

0.32mins
Q1e

Graph each functions given f(x) = -x + 2 and g(x) = (x + 3)^2.

a) y = f(g(x))

b) y = g(f(x))

c) y = f(f(x))

d) y = g(g(x))

e) y = f^{-1}(f(x))

2.23mins
Q2

Let f(x) = x^2 + 2x -4 and g(x) = \frac{1}{x + 1}.

Evaluate g(f(0))

0.26mins
Q4a

Let f(x) = x^2 + 2x -4 and g(x) = \frac{1}{x + 1}.

Evaluate f(g(-2))

0.21mins
Q4b

Let f(x) = x^2 + 2x -4 and g(x) = \frac{1}{x + 1}.

Show that g(f(x)) is undefined for x = 1 and x = -3.

0.35mins
Q4c

Is f(g(x)) = g(f(x)) true for all functions f(x) and g(x)? Show your work.

1.01mins
Q8

Let f(x) = x^3.

a) Determine f^{-1}(x).

b) Determine f(f^{-1}(x)).

c) Determine f^{-1}(f(x)).

d) Compare your answer to parts b) and c). Describe what you notice.

e) Determine f(f^{-1}(3)), f(f^{-1}(5)), and f(f^{-1}(-1)). What do you notice.

1.28mins
Q9

Is (f \circ g)^{-1} = (f^{-1}\circ g^{-1})(x), or is (f \circ g)^{-1}(x) = (g^{-1}\circ f^{-1})(x), or is neither true? Verify using examples.

2.32mins
Q14

Let f(x) = x^2, g(x) = x - 2, and h(x) = \frac{1}{x}.

a) Determine a simplified algebraic model for each composite function.

• i. f(g(x))
• ii. h(g(x))
• iii. g^{-1}(h(x))

b) Evaluate f(h(2))

1.02mins
Q17

Let f(x) = x^2 - 9 and g(x) = \frac{1}{x}.

• Determine the domain and range of y = f(g(x))
2.00mins
Q19a

Let f(x) = x^2 - 9 and g(x) = \frac{1}{x}.

• Determine the domain and range of y = g(f(x))
1.43mins
Q19b

Given that f(x) =\dfrac{ax + b}{cx + d}, find f^{-1}(x).

1.12mins
Q21

Find all real numbers x such that \sqrt{1 - \sqrt{1 - x}} = x