College Algebra Exam 3 Practice: Polynomial, Rational, Exponential, and Logarithmic Functions

5.1 Polynomial Functions

Transformations and Graphing

Polynomial functions are algebraic expressions involving powers of x with real coefficients. Transformations such as shifting, stretching, and reflecting are used to graph these functions.

• Key Point: The general form of a polynomial function is $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$.

• Example: For $f(x) = x^2 - (x - 1)^2$, label three points on the graph to illustrate transformations.

• Key Point: Multiplicity of zeros affects the graph's behavior at those points. For example, a zero with multiplicity two will touch the x-axis but not cross it.

• Example: Find a fourth degree polynomial with zeros at $-2$, $1$, and $4$, where the zero at $1$has multiplicity two. The function is$f(x) = a(x + 2)(x - 1)^2(x - 4)$.

5.2 Graphing Polynomial Functions; Models

Analyzing Polynomial Graphs

Graphing polynomial functions involves identifying intercepts, intervals of increase/decrease, and behavior at zeros and asymptotes.

• Key Point: The end behavior of a polynomial is determined by its leading term.

• Key Point: Intervals where the graph is above or below the x-axis can be found by solving $f(x) > 0$ or $f(x) < 0$.

• Example: For $f(x) = x^3 - 7x^2 - 5x^2 + 3x^4 + 4x^3$, determine intercepts, intervals, and graph behavior.

5.3 Properties of Rational Functions

Overview

Rational functions are quotients of polynomials. Their properties include domain, intercepts, and asymptotes.

• Key Point: The domain excludes values that make the denominator zero.

5.4 The Graph of a Rational Function

Graphing and Analyzing Rational Functions

Rational functions can have vertical, horizontal, or oblique asymptotes. Their graphs are analyzed for intercepts, domain, and behavior near asymptotes.

• Key Point: The general form is $R(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.

• Example: For $R(x) = \frac{x}{x-2}$, find domain, intercepts, intervals above/below x-axis, and asymptotes.

• Key Point: Vertical asymptotes occur at zeros of the denominator; horizontal asymptotes depend on the degrees of numerator and denominator.

• Example: For $R(x) = \frac{x^2 - 12}{x^2 - 1}$, analyze all properties and graph.

5.5 Polynomial and Rational Inequalities

Solving Inequalities

Polynomial and rational inequalities involve finding the set of values for which the function is positive or negative.

• Key Point: The domain of a function like $g(x) = \sqrt{\frac{x}{x+2}}$ is found by ensuring the radicand is non-negative and the denominator is not zero.

5.6 The Real Zeros of a Polynomial Function

Finding Real Zeros

Real zeros of a polynomial are the x-values where the function equals zero. These can be found by factoring or using the Rational Root Theorem.

• Key Point: The Fundamental Theorem of Algebra states that a polynomial of degree n has n complex zeros (counting multiplicities).

5.7 Complex Zeros; Fundamental Theorem of Algebra

Complex Zeros and Factored Form

Polynomials with real coefficients may have complex zeros. These are found using factoring, completing the square, or the quadratic formula.

• Example: For $f(x) = x^5 + x^4 + x^3 + x^2 + x - 175$, find all complex zeros and write in factored form.

6.1 Composite Functions

Composition and Domain

Composite functions are formed by applying one function to the result of another. The domain of the composite is restricted by both functions.

• Key Point: If $f(x)$ and $g(x)$ are functions, then $(f \circ g)(x) = f(g(x))$.

• Example: If $f(x) = \frac{x}{x-1}$ and $g(x) = \sqrt{x}$, find $f(g(x))$ and $g(f(x))$, stating the domain for each.

6.2 One-to-One Functions; Inverse Functions

Finding and Verifying Inverses

One-to-one functions have unique outputs for each input and possess inverses. The inverse function reverses the effect of the original function.

• Key Point: If $f(x)$ is one-to-one, then $f^{-1}(x)$ exists and $f(f^{-1}(x)) = x$.

• Example: For $f(x) = x^3 + e^x$, find the domain, range, and inverse.

6.3 Exponential Functions

Solving Exponential Equations

Exponential functions have the form $f(x) = a^x$. Equations involving exponentials are solved using logarithms or by equating exponents.

• Example: Solve $11^{x+8} = 12^x$.

6.4 Logarithmic Functions

Properties and Graphing

Logarithmic functions are the inverses of exponential functions. Their properties include domain, range, and asymptotes.

• Key Point: The logarithm function $f(x) = \log_b(x)$ is defined for $x > 0$.

• Example: Find the exact value of $\log_{1/2} 1/32$.

• Key Point: The domain of $f(x) = \log_2(x - 1)$ is $x > 1$.

6.5 Properties of Logarithms

Simplifying Logarithmic Expressions

Logarithmic properties allow expressions to be simplified, expanded, or written as a single logarithm.

• Key Point: $\log_b(MN) = \log_b(M) + \log_b(N)$

• Key Point: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

• Key Point: $\log_b(M^k) = k\log_b(M)$

• Example: Simplify $\log_7 7 + \log_7 16$.

6.6 Logarithmic and Exponential Equations

Solving Equations

Equations involving logarithms and exponentials are solved by applying properties of logarithms and exponentials, and sometimes by rewriting in equivalent forms.

• Example: Solve $\log_2(x + 1) + \log_2(x - 1) = 2$.

• Example: Solve $\left(\frac{1}{2}\right)^{x^2 - 2x} = 2^{-x}$.