Composite and Inverse Functions: Precalculus Study Notes

Composite Functions

Definition and Formation of Composite Functions

Composite functions are formed when the output of one function becomes the input of another. If f and g are functions, the composite function f ∘ g is defined as (f ∘ g)(x) = f(g(x)). The domain of f ∘ g consists of all x in the domain of g such that g(x) is in the domain of f.

• Key Point 1: The notation f ∘ g means "f composed with g" or "f of g of x".

• Key Point 2: The domain of f ∘ g is restricted by both the domain of g and the requirement that g(x) must be in the domain of f.

• Example: If f(x) = 5x - 1 and g(x) = 3x^2, then (f ∘ g)(1) = f(g(1)) = f(3) = 5(3) - 1 = 14.

Evaluating Composite Functions

To evaluate a composite function at a specific value, substitute the value into the inner function, then use the result as the input for the outer function.

• Example: Given f(x) = 5x - 1 and g(x) = 3x^2:

  • (f ∘ g)(1) = f(g(1)) = f(3) = 14

  • (g ∘ f)(1) = g(f(1)) = g(4) = 3(4^2) = 48

  • (f ∘ f)(-2) = f(f(-2)) = f(-11) = 5(-11) - 1 = -56

  • (g ∘ g)(-2) = g(g(-2)) = g(12) = 3(12^2) = 432

Composite Functions with Radicals

When composite functions involve radicals, ensure the input values are within the domain of the radical function (i.e., the radicand is non-negative for even roots).

• Example: If f(x) = 6\sqrt{x} and g(x) = 2x, then (f ∘ g)(4) = f(g(4)) = f(8) = 6\sqrt{8} = 12\sqrt{2}.

Finding the Domain of a Composite Function

To determine the domain of (f ∘ g)(x), exclude any x not in the domain of g and any x for which g(x) is not in the domain of f.

• Example: If f(x) = x^2 + 2x - 5 and g(x) = 3x + 1, both have domain all real numbers, so f ∘ g and g ∘ f also have domain all real numbers.

• Example: If f(x) = \frac{2}{x+4} and g(x) = \frac{4}{x-3}, then the domain of g is x ≠ 3. For f ∘ g, also exclude x such that g(x) = -4, which leads to further restrictions.

Summary Table: Domain Restrictions in Composite Functions

Step	Restriction
1. Domain of g	Exclude values not in domain of g
2. Domain of f ∘ g	Exclude values where g(x) is not in domain of f

One-to-One and Inverse Functions

One-to-One Functions

A function is one-to-one if every output is paired with exactly one input. That is, if f(x_1) = f(x_2) implies x_1 = x_2.

• Key Point 1: One-to-one functions pass the horizontal-line test: every horizontal line intersects the graph at most once.

• Key Point 2: If a function is strictly increasing or decreasing on an interval, it is one-to-one on that interval.

• Example: f(x) = x^5 is one-to-one, but f(x) = x^2 is not.

Inverse Functions

If f is a one-to-one function, its inverse function f^{-1} reverses the mapping of f. That is, f^{-1}(f(x)) = x and f(f^{-1}(x)) = x for all x in the appropriate domains.

• Key Point 1: The domain of f is the range of f^{-1}, and vice versa.

• Key Point 2: The graph of f^{-1} is the reflection of the graph of f across the line y = x.

• Example: If f(x) = 3x - 5, then f^{-1}(x) = \frac{x + 5}{3}.

Finding the Inverse of a Function

To find the inverse of a one-to-one function defined by an equation:

1. Replace f(x) with y.

2. Interchange x and y.

3. Solve for y in terms of x; this is f^{-1}(x).

4. Verify by checking f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

• Example: For f(x) = \frac{1}{x+4}, interchange variables to get x = \frac{1}{y+4}, then solve for y to find the inverse.

Verifying Inverse Functions

To verify that two functions are inverses, show that composing one with the other yields the identity function:

• f(f^{-1}(x)) = x for all x in the domain of f^{-1}

• f^{-1}(f(x)) = x for all x in the domain of f

Summary Table: Properties of Inverse Functions

Property	Description
Domain of f	Range of f^{-1}
Range of f	Domain of f^{-1}
f(f^{-1}(x))	x
f^{-1}(f(x))	x

Additional info:

• Composite functions are often illustrated with diagrams showing the flow from the domain of g to the range of f.

• One-to-one functions are essential for the existence of inverse functions.

• When working with rational or radical functions, always check for domain restrictions due to division by zero or even roots of negative numbers.