Overview
This lecture reviewed key concepts in functions, including function evaluation, inverse functions, composition, graphs, one-to-one functions, and domain restrictions.
Function Evaluation and Inverse Functions
- To evaluate a function at a value, substitute the value for x (e.g., f(2) = 2³ + 5×2² − 9 = 19).
- To find an inverse, substitute f(x) with y, switch x and y, solve for y, and simplify.
- For radical inverses, raise both sides to eliminate the root and isolate y.
Solving for x Given f(x)
- Set the function equal to the given value and rearrange all terms to one side.
- Factor the resulting quadratic/trinomial expression as needed.
- Set each factor to zero and solve for x.
Function Composition
- To compute f(g(x)), substitute g(x) into each x of f and expand/simplify.
- Combine like terms for the final simplified expression.
Composition with Function Values
- Find the inner function value (e.g., f(2)), use its result as input for the outer function (g(f(2))).
Inverse Functions Verification
- Two functions are inverses if f(g(x)) = x and g(f(x)) = x when simplified.
- Substitute one function into the other and simplify to check.
Graphs: Functions and One-to-One
- A graph is a function if it passes the vertical line test (each x-value corresponds to one y).
- A function is one-to-one if it passes the horizontal line test (each y-value corresponds to one x).
- Non-one-to-one functions don’t have inverses that are functions.
Solving for Variables in Function Output
- Substitute known values, simplify, and factor to solve for the unknown (e.g., find y when f(4, y) = 48).
Domain of Rational Functions
- The domain excludes values making the denominator zero.
- Factor the denominator, set factors to zero, solve for x, and exclude those x-values.
- Express the domain using interval notation, excluding restricted values.
Key Terms & Definitions
- Function — a rule assigning each input exactly one output.
- Inverse Function — a function that reverses the effect of another function.
- Domain — all possible input values for a function.
- Vertical Line Test — a test to determine if a graph is a function.
- Horizontal Line Test — a test to determine if a function is one-to-one.
Action Items / Next Steps
- Practice evaluating, composing, and inverting functions.
- Review factoring quadratics and trinomials.
- Complete any assigned problems on function domains and graph identification.