Identifying parts of Composite Functions
Composition using set of points
Domain of Composite Function
Domain of Composite Function
Domain of Composite Function, challenging
Domain of Composite Function ex2
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.
Odd Function
Composition with algegra
Decomposition
Triple composition
Decomposition example
Decomposition example 3
Sketching degree 1 Rational Function
Let f(x) = -x + 2
and g(x) = (x + 3)^2
. Determine a simplified algebraic model for the composite function.
y = f(g(x))
Let f(x) = -x + 2
and g(x) = (x + 3)^2
. Determine a simplified algebraic model for the composite function.
y = g(f(x))
Let f(x) = -x + 2
and g(x) = (x + 3)^2
. Determine a simplified algebraic model for the composite function.
y = f(f(x))
Let f(x) = -x + 2
and g(x) = (x + 3)^2
. Determine a simplified algebraic model for the composite function.
y = g(g(x))
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))
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))
Let f(x) = x^2 + 2x -4
and g(x) = \frac{1}{x + 1}
.
Evaluate g(f(0))
Let f(x) = x^2 + 2x -4
and g(x) = \frac{1}{x + 1}
.
Evaluate f(g(-2))
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
.
Is f(g(x)) = g(f(x))
true for all functions f(x)
and g(x)
? Show your work.
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.
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.
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.
f(g(x))
h(g(x))
g^{-1}(h(x))
b) Evaluate f(h(2))
Let f(x) = x^2 - 9
and g(x) = \frac{1}{x}
.
y = f(g(x))
Let f(x) = x^2 - 9
and g(x) = \frac{1}{x}
.
y = g(f(x))
Given that f(x) =\dfrac{ax + b}{cx + d}
, find f^{-1}(x)
.
Find all real numbers x
such that \sqrt{1 - \sqrt{1 - x}} = x
The sum of two numbers is 7 and their product is 25 . Determine the sum of their reciprocals.