Overview
This lecture covers composite functions, one-to-one functions, and inverse functions, explaining key definitions, notations, examples, and necessary domain restrictions.
Composite Functions
- A composite function is written as f ∘ g or f(g(x)), meaning f composed with g.
- The inner function is evaluated first, then its output becomes the input for the outer function.
- The domain of a composite function is all x-values in the domain of g such that g(x) is in the domain of f.
- Example: If H(x) = 2x² – 3 and J(x) = 4x, then H(J(1)) = 29.
- Order matters: H(J(1)) ≠ J(H(1)).
- Composite functions can be formed with variables: H(J(x)) = 32x² – 3, J(H(x)) = 8x² – 12.
Constructing Individual Functions
- Given H(x) = √(x³ + 5), identify g(x) = x³ + 5 (inner) and f(x) = √x (outer).
- Alternatively, find the “powered” variable to identify g(x), and substitute back into f(x).
- Only one valid set of f and g is needed, though there may be multiple ways.
One-to-One and Inverse Functions
- A function is one-to-one if every x-value maps to a unique y-value, and vice versa.
- One-to-one functions pass both vertical and horizontal line tests on their graphs.
- The inverse of a function, denoted f⁻¹(x), switches the roles of x and y.
- To verify inverses, show f(g(x)) = x or g(f(x)) = x.
- For sets of points, swap x and y values to form the inverse.
Finding Inverse Functions
- For equations, replace f(x) with y, switch x and y, then solve for y.
- Example: For f(x) = 3x + 5, inverse is f⁻¹(x) = (x – 5)/3.
- For g(x) = (3/4)x + 1, the inverse is g⁻¹(x) = (4/3)(x + 1).
- For nonlinear functions such as cube roots and square roots, follow similar steps but watch for plus/minus and domain restrictions.
Domain Restrictions for Inverses
- Some functions (e.g., parabolas) are not one-to-one unless domain is restricted.
- For f(x) = √(x + 5) + 3, restrict domain to x ≥ –3 to ensure one-to-one.
- Use inequalities for clarity: for positive square roots, x ≥ vertex x-coordinate; for negative, x ≤ vertex x-coordinate.
Key Terms & Definitions
- Composite Function — Combining two functions such that the output of one becomes the input of the other.
- One-to-One Function — A function where each input maps to exactly one unique output, and vice versa.
- Inverse Function — A function that reverses the original function, denoted f⁻¹(x).
- Domain Restriction — Limiting the input values to ensure a function is one-to-one.
Action Items / Next Steps
- Practice finding composite and inverse functions, including with domain restrictions.
- Memorize the process for restricting domains for functions involving square roots or parabolas.
- Complete assigned problems on verifying and finding inverse functions as given by your instructor.