Polynomial, Rational, and Composite Functions: Study Guide and Practice Problems

Polynomial and Rational Functions

Overview

This section covers the analysis and manipulation of polynomial and rational functions, including their domains, intercepts, asymptotes, and zeros. Understanding these concepts is essential for graphing and solving equations involving these functions.

Polynomial Functions

• Definition: A polynomial function is an expression of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer.

• Degree: The highest power of $x$ in the polynomial.

• Zeros: Values of $x$ for which $f(x) = 0$.

• Multiplicity: The number of times a particular zero occurs.

• End Behavior: Determined by the leading term $a_nx^n$.

• Example: $f(x) = x^3 - 4x^2 + 7x - 6$ is a cubic polynomial.

Rational Functions

• Definition: A rational function is a ratio of two polynomials: $f(x) = \frac{p(x)}{q(x)}$, where $q(x) \neq 0$.

• Domain: All real numbers except where $q(x) = 0$.

• Vertical Asymptotes: Occur at zeros of $q(x)$ that are not canceled by zeros of $p(x)$.

• Horizontal Asymptotes: Determined by the degrees of $p(x)$ and $q(x)$.

• Oblique Asymptotes: Occur when the degree of $p(x)$ is exactly one more than the degree of $q(x)$.

• Holes: Occur at values canceled by both numerator and denominator.

• Example: $f(x) = \frac{x^2-4}{x-2}$ has a hole at $x=2$.

Analyzing Graphs of Rational Functions

1. Find the domain of the function.

2. Write the function in lowest terms.

3. Find the x-intercept(s) and y-intercept.

4. Find vertical asymptotes.

5. Find horizontal or oblique asymptotes.

6. Determine where the graph is above or below the x-axis.

Composite Functions

• Definition: The composition of functions $f$ and $g$ is $(f \circ g)(x) = f(g(x))$.

• Domain: The set of all $x$ such that $x$ is in the domain of $g$ and $g(x)$ is in the domain of $f$.

• Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f \circ g)(x) = (x+1)^2$.

Inverse Functions

• Definition: The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$, so $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Finding the Inverse: Solve $y = f(x)$ for $x$ in terms of $y$, then interchange $x$ and $y$.

• Example: If $f(x) = 2x+3$, then $f^{-1}(x) = \frac{x-3}{2}$.

Solving Equations and Inequalities

Polynomial Equations

• Solving: Set the polynomial equal to zero and factor or use the quadratic formula for degree 2 polynomials.

• Example: $x^2 - 4x + 3 = 0$ has solutions $x = 1$ and $x = 3$.

Rational Equations

• Solving: Multiply both sides by the least common denominator to clear fractions, then solve the resulting equation.

• Check for extraneous solutions: Solutions that make any denominator zero must be excluded.

Polynomial and Rational Inequalities

• Solving: Find the zeros of the numerator and denominator, plot them on a number line, and test intervals.

• Express the solution in interval notation.

• Example: Solve $\frac{x^2-4}{x-2} > 0$.

Graphing Functions

Steps for Graphing

1. Find the domain.

2. Find intercepts.

3. Find asymptotes.

4. Plot key points and sketch the graph.

5. Determine the range.

Tables: Properties and Comparisons

Sample Table: Function Values

x	f(x)	g(x)
1	2	4
2	3	6
3	5	10
4	6	4

Key Formulas and Theorems

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Intermediate Value Theorem: If $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then there is at least one $c$ in $(a, b)$ such that $f(c) = 0$.

• Bounds on Zeros Theorem: Provides upper and lower bounds for the real zeros of a polynomial.

Practice Problems

• Find the domain, intercepts, and range of piecewise and rational functions.

• Graph polynomial and rational functions, identifying asymptotes and holes.

• Solve equations and inequalities, expressing answers in interval notation.

• Analyze composite and inverse functions, including their domains.

• Use tables to evaluate composite functions.

Additional info:

• Some problems require using the Intermediate Value Theorem and Bounds on Zeros Theorem for polynomials.

• Graphical analysis includes identifying holes, intercepts, and asymptotes for rational functions.

• Practice includes both real and complex solutions for equations.