Chapter 10: Compositions, Inverses, and Combinations of Functions

10.1 Composition of Functions

The composition of functions is a fundamental concept in precalculus, allowing the output of one function to become the input of another. This process is essential for modeling complex relationships and understanding how functions interact.

• Composite Function: A composite function, denoted as , uses the output of the inner function as the input to the outer function .

• Domain Rule: The composite is defined only for -values in the domain of whose outputs are within the domain of .

• Decomposition: The reverse process involves breaking down a complex function into simpler component functions and such that .

• Key Skills:

  • Evaluating compositions using formulas, graphs, or tables.

  • Finding composite formulas (e.g., given and , find ).

  • Identifying inner and outer functions from a combined formula.

• Example: If and , then .

10.2 Revisiting Inverse Functions

Inverse functions reverse the effect of a function, mapping outputs back to their original inputs. Understanding inverses is crucial for solving equations and analyzing function behavior.

• Definition: if and only if . The output units of match the input units of .

• Horizontal Line Test (HLT):

  • If any horizontal line crosses a function's graph more than once, the function is not invertible.

  • If every horizontal line intersects at most once, the function has an inverse.

• Finding Formulas for Inverses: Set , solve for in terms of , then swap variables so $x$ is the independent variable.

• Example: For , set , solve for : , so .

10.3 The Graph, Domain, and Range of an Inverse Function

The graph of an inverse function is a reflection across the line . The domain and range of a function and its inverse are swapped, and fundamental identities relate the two.

• Graphical Reflection: The graph of is a reflection of across the line . If lies on , then lies on .

• Domain and Range Swap:

  • Domain of = Range of

  • Range of = Domain of

• Fundamental Identities:

  • for all in domain of

  • for all in domain of

• Restricting Domain: Functions failing the HLT (like ) can be restricted (e.g., ) to create an invertible piece.

• Example: The function is not invertible on , but is invertible on .

10.4 Combinations of Functions

Functions can be combined through addition, subtraction, multiplication, and division. These operations are used to model and interpret real-world scenarios.

• Combining Operations:

  • Sum:

  • Difference:

  • Product:

  • Quotient: (where )

• Practical Applications:

  • Food Surplus:

  • Per-Capita Rates: (e.g., food supply per person or crime rate per person).

• Example: If is the total food supply and is the population, then gives the food supply per person.

Chapter 11: Polynomial and Rational Functions

11.1 Power Functions and Proportionality

Power functions are a class of functions defined by an exponent, and their behavior varies depending on the value of the exponent. Proportionality describes relationships where one quantity varies directly or inversely with a power of another.

• Power Function Definition: where and are constants.

• Behavior by Power (p):

  • Positive integers (): Pass through origin; degree determines end behavior.

  • Negative integers (): Unbounded near ; approach 0 as .

  • Fractional powers (): Represent root functions.

• Proportionality:

  • is directly proportional to if

  • is inversely proportional if

• Example: is a power function with ; is inversely proportional to .

11.2 Polynomial Functions and Their Behavior

Polynomial functions are sums of power functions with non-negative integer exponents. Their behavior is determined by their degree and coefficients.

• General Formula:

• Long-Run Behavior: As , behaves like its leading term .

• Short-Run Behavior: Governed by its roots/zeros and turning points.

• Example: is a cubic polynomial.

11.3 Zeros of Polynomials and Short-Run Behavior

The zeros of a polynomial are the values of where . The behavior at each zero depends on its multiplicity, and the number of turning points is related to the degree.

• Factored Form:

• Multiplicity of Zeros:

  • Odd multiplicity (1, 3, ...): The graph crosses the x-axis at the zero.

  • Even multiplicity (2, 4, ...): The graph touches and turns around at the zero.

• Turning Points: A polynomial of degree has at most turning points.

• Example: has a zero at (multiplicity 2) and (multiplicity 1).

11.4 & 11.5 Rational Functions & Their Short-Run Behavior

Rational functions are quotients of polynomials. Their behavior is characterized by zeros, vertical and horizontal asymptotes, and removable discontinuities (holes).

• Definition: , where and are polynomials.

• Vertical Asymptotes: Occur where (and ).

• Zeros: Occur where (and ).

• Horizontal Asymptotes (Long-Run Behavior):

  • Degree of Degree of

  • Degree of Degree of

  • Degree of Degree of No horizontal asymptote

• Holes (Removable Discontinuities): Occur if a factor cancels out completely from both and .

• Example: has a hole at and a vertical asymptote at .