Composite Functions: Formation, Evaluation, and Domain

Composite Functions

Definition of Composite Function

A composite function is formed when one function is applied to the result of another function. Given two functions f and g, the composite function f ˆ g (read as "f composed with g") is defined as:

• Notation: $(f ext{ o } g)(x) = f(g(x))$

• This means you first apply g to x, then apply f to the result.

• The domain of f ˆ g is all values of x in the domain of g for which g(x) is in the domain of f.

Example: If $f(x) = 2x^2 - 3$ and $g(x) = 4x$, then $f(g(x)) = 2(4x)^2 - 3 = 32x^2 - 3$.

Evaluating Composite Functions

Step-by-Step Evaluation

To evaluate a composite function at a specific value:

1. Find the value of the inner function g(x) at the given input.

2. Use this result as the input for the outer function f.

Example 1: Suppose $f(x) = 2x^2 - 3$ and $g(x) = 4x$.

• Find $f(g(1))$: $g(1) = 4 imes 1 = 4$ $f(4) = 2 imes 4^2 - 3 = 2 imes 16 - 3 = 32 - 3 = 29$

• Find $g(f(-1))$: $f(-1) = 2 imes (-1)^2 - 3 = 2 imes 1 - 3 = -1$ $g(-1) = 4 imes (-1) = -4$

Composite Functions with Rational and Radical Functions

When working with rational or radical functions, pay special attention to the domain restrictions.

• Example: $f(x) = rac{8}{x+2}$, $g(x) = rac{3}{ ext{sqrt}(x)}$

• Find $f(g(4))$: $g(4) = rac{3}{2}$ $figg( rac{3}{2}igg) = rac{8}{ rac{3}{2} + 2} = rac{8}{ rac{7}{2}} = rac{16}{7}$

Composite Functions Using Tables and Graphs

Composite functions can also be evaluated using tabular or graphical data.

• Use the table to find the value of the inner function, then use that value to find the outer function.

x	f(x)	g(x)
-3	2	4
-2	4	2
-1	0	3
0	1	0
1	2	1
2	3	2
3	4	3

Example: To find $f(g(1))$, first find $g(1) = 1$, then $f(1) = 2$.

Finding the Domain of Composite Functions

General Principles

When determining the domain of $f ext{ o } g$, consider:

• 1. Any $x$ not in the domain of $g$ must be excluded.

• 2. Any $x$ for which $g(x)$ is not in the domain of $f$ must be excluded.

Example 2: Polynomial Functions

Suppose $f(x) = x^2 + 3x - 1$ and $g(x) = 2x + 3$.

• $f(g(x)) = (2x+3)^2 + 3(2x+3) - 1 = 4x^2 + 12x + 9 + 6x + 9 - 1 = 4x^2 + 18x + 17$

• Domain: $(-0, 0)$ (all real numbers, since polynomials are defined everywhere)

Example 3: Rational Functions

Suppose $f(x) = rac{1}{x+2}$ and $g(x) = rac{4}{x-1}$.

• Find the domain of $f ext{ o } g$:

• First, exclude values not in the domain of $g$: $x eq 1$

• Next, exclude values for which $g(x)$ is not in the domain of $f$: $g(x) eq -2$

• Solve $g(x) = -2$: $rac{4}{x-1} = -2 ightarrow x = -1$

• Domain: $x eq 1, x eq -1$

Example 4: Radical Functions

Suppose $f(x) = ext{sqrt}(x)$ and $g(x) = 2x + 5$.

• $f(g(x)) = ext{sqrt}(2x+5)$

• Domain: $2x+5  0 ightarrow x  -2.5$

Summary Table: Domain Restrictions in Composite Functions

Step	Restriction
1	$x$ must be in the domain of $g$
2	$g(x)$ must be in the domain of $f$

Key Points

• Composite functions combine two functions by applying one to the result of the other.

• To evaluate a composite function, work from the inside out.

• The domain of a composite function is restricted by both the inner and outer functions.

• Composite functions can be evaluated using formulas, tables, or graphs.

Additional info:

• When working with rational or radical functions, always check for values that make denominators zero or radicands negative.

• Composite functions are foundational for understanding more advanced topics such as function transformations and inverses.