6.5 Chapter 6 Study Guide

Definitions

• rational function: a function that is a quotient of two polynomials, \(f(x) = \dfrac{p(x)}{q(x)} \).
• vertical asymptote: a vertical line with equation \(x = a\) that a function approaches infinitely but never touches. As \(x\) approaches \(a\) from one side or the other, the function shoots off to positive or negative infinity, hugging the vertical line.
• horizontal asymptote: a horizontal line with equation \( y = a\) that a function approaches as \(x \rightarrow \infty\) and/or \( x \rightarrow -\infty\).
• hole: a location where a rational expression is not defined due to a domain issue, but doesn't have a V.A.
• slant asymptote: a line (linear function) that a rational function "behaves like" as \(x \rightarrow \infty\). Only happens if the degree of the numerator is exactly one more than the degree of the denominator.
• limit of \(f\) at \( x = a\): the \(y\)-value that the function's graph is approaching as \(x\) approaches \(a\) from both the left and the right.
• one-sided limit: a limit where you only approach \(x = a\) from the left or the right.
• exponential function: a function of the form \(f(x) = a^x\), with \(a>0, a\neq 1\). The number \(a\) is the base.
• logarithmic function: the logarithmic function with base \(a\), \( \log_a x\), is defined by \( \log_a x = y \quad \iff \quad a^y = x \).
• natural log: the natural log function \(\ln x\) is the logarithmic function with base \(e\).

Facts About Rational Functions

• Their domain is all real numbers except any values of \(x\) that cause \(q(x)\) to be zero. You can write this in set notation as \( \{ x \: | \: q(x) \neq 0 \} \).
• Often, your first move should be to fully factor the top and bottom. Then to identify...
  • the domain of a rational function, just look at the denominator alone, set it equal to 0, and solve for \(x\). Those values cause division by zero, so they cannot be allowed in the domain. Those values also give the vertical asymptotes, which will have equations of the form \(x = c\).
  • the roots (\(x\)-intercepts) of a rational function, just look at the numerator alone, set it equal to 0, and solve for \(x\). Aka, the roots of the numerator give the roots of the rational function.
  • any horizontal asymptotes of a rational function using the graph, look for an invisible horizontal line that the function hugs as \(x \rightarrow \infty\) or \(x \rightarrow -\infty\). If the height of that line is \(c\), then the equation of the horizontal asymptote is \(y = c\).

Partial Fraction Decomposition

To determine \(A, B,\) and \(C\) such that \( f(x) = \dfrac{ x+1}{x(x-1)(x+2)} = \dfrac{A}{x} + \dfrac{B}{x-1} + \dfrac{C}{x+2} \),

1. Multiply through the entire equation by the left hand side's whole denominator to clear all denominators.
2. Compare coefficients of like terms on both sides of the equation. Each power of \(x\) will give an equation involving \(A, B,\) and \(C\).
3. Solve the resulting system of equations to find values for \(A, B,\) and \(C\).

Solving a Polynomial Inequality

1. Move all the terms to one side and get 0 on the other.
2. Factor the polynomial into linear factors and irreducible quadratic factors, and find the real roots.
3. Make a number line, divided into zones (intervals) by the roots.
4. In each zone, determine whether the function is positive or negative by using logic or testing points and comparing the signs of the factors.
5. Report the desired zones as intervals, based on the inequality sign.

Solving a Rational Inequality

1. Move all the terms to one side into a single giant fraction and get 0 on the other.
2. Factor the numerator and the denominator. Identify the points of interest where either the numerator or the denominator are zero. (Basically, roots of the top poly and roots of the bottom poly.)
3. Make a number line, divided into zones (intervals) by the points of interest.
4. In each zone, determine whether the function is positive or negative by using logic or testing points and comparing the signs of the factors.
5. Report the desired zones as intervals, based on the inequality sign. Test the endpoints carefully to see if they should be included, because the function may not be defined at some of them!

Limit of a Function

When looking at a graph of \(y = f(x) \), the limit of \(f\) at \(x = a\) is the \(y\)-value that the function appears to be heading toward as you move towards a particular \(x\)-value of interest, \(a\), from both the left and the right. The existence of a limit does depend on the function actually approaching a value from both directions.

In other words, nicknaming the limit "\(L\)" if it exists, we're saying that as \(x \rightarrow a \), \(f(x) \rightarrow L \), in the arrow notation we've used before. NOTE: the limit does not care what the actual function value \(f(a)\) is, or even whether it's defined! It's all about where the function ACTS like it's going.

We use the limit notation

\[ \lim_{x \rightarrow a} f(x) = L \notag \]

if the limit exists properly. If a function tends toward \( \pm \infty \) when approaching \( x = a \) from both directions, the limit is infinite, and although \(\infty\) is not a number, we still use the notation

\[ \lim_{x \rightarrow a} f(x) = (\pm) \infty \notag \]

Sometimes we are interested in one-sided limits, looking at the behavior of the function as we come in from only one side. In this case, we use a "\(^+\)" decoration to indicate approaching from the right, and a "\(^-\)" to indicate approaching from the left.

\[ \text{approaching \(x = a\) from the right only: } \lim_{x \rightarrow a^+} f(x) = L, \quad \text{ approaching from the left only: } \lim_{x \rightarrow a^- } f(x) = L \notag \]

Limit Examples

The limit exists and agrees with the function value at that location.	The limit exists even though the function is not defined at that location.
The one-sided limits exist, but do not agree, so the limit as a whole does not exist.	The one-sided limits are infinite, but do not agree, so the limit as a whole does not exist.
The function goes up forever, so the "limit at infinity" is infinite.	The function goes down forever, so the "limit at negative infinity" is negative infinity.
The one-sided limits are infinite, but they do agree, so we say the limit is infinity.	The "limit at infinity" is leveling out, hugging a certain horizontal line, so it is finite.

Vertical Asymptotes via Limits

The vertical line \( x = a\) is a vertical asymptote of a function \( f(x) \) if ANY of the following are true:

\[ \lim_{x \rightarrow a} f(x) = \infty \notag \]	\[ \lim_{x \rightarrow a^+} f(x) = \infty \notag \]	\[ \lim_{x \rightarrow a^-} f(x) = \infty \notag \]
\[ \lim_{x \rightarrow a} f(x) = -\infty \notag \]	\[ \lim_{x \rightarrow a^+} f(x) = -\infty \notag \]	\[ \lim_{x \rightarrow a^-} f(x) = -\infty \notag \]

Horizontal Asymptotes via Limits

The horizontal line \(y = L \) is a horizontal asymptote of a function \(f(x)\) if

\[ \lim_{x \rightarrow \infty} f(x) = L \quad \text{ and/or } \quad \lim_{x \rightarrow -\infty} f(x) = L \notag \]

Finding a Slant Asymptote

1. Make sure you're looking at a rational function \( \dfrac{p(x)}{d(x)} \) where the degree of \(p(x)\) is exactly one more than the degree of \(d(x)\).
2. Perform polynomial long division, writing \( \dfrac{p(x)}{d(x)} = q(x) + \dfrac{r(x)}{d(x)} \).
3. The slant asymptote is \( y = q(x) \).

Facts About Exponential Functions

Exponential functions are functions of the form \( f(x) = a^x\) for some base \(a > 0, a \neq 1\).

• In an exponential function, the base is a plain ol' number and the variable is in the exponent. Not to be confused with polynomial expressions where the variable is the base and a number is the power, like \(x^3\)! Whole different animal.
• The domain of an exponential function is all real numbers.
• If \(a > 1\), the function is increasing (graph going uphill from left to right). If \( 0 < a< 1\), the function is decreasing (graph going downhill from left to right).
• End behavior for \(a > 1\): As \(x \rightarrow -\infty\), the graph hugs a horizontal asymptote at the \(x\)-axis. As \(x \rightarrow \infty\), the graph shoots up rapidly, and the bigger the base, the faster the growth.
• End behavior for \( 0<a<1\): As \(x \rightarrow -\infty\), the graph is blowing up to the left. As \(x \rightarrow \infty\), the graph hugs the horizontal asymptote at the \(x\)-axis forever.
• All exponential functions pass through \( (0,1)\) as well as \( (1,a)\), for whatever their base \(a\) is.
• The function values are always positive, aka above the \(x\)-axis, aka the range is \( (0, \infty) \).
• For \( f(x) = a^x\) and \( g(x) = \left( \frac{1}{a}\right)^x \), note that \( g(x) = \left( \frac{1}{a}\right)^x = \frac{1}{a^x} = a^{-x} = f(-x) \).

Facts About Logarithmic Functions

The logarithmic function with base \(a\) (\(a > 0, a \neq 1\)), \( \log_a x\), is defined by

\[ \log_a x = y \quad \iff \quad a^y = x. \notag \]

• Basically, "\( \log_a (x) = ?\)" is asking the question, "What do I need to raise the base \(a\) to in order to get out this \(x\)?" The answer to that question is the output of the function \( \log_a\).
• When in particular the base is the special number \(e\), we write \( \log_e x \) as \( \ln x\), and call it the natural log of \(x\).
• Their domain is only positive real numbers! You are not allowed to put 0 or a negative number into a log, because that would be the same as saying \( a^{y} = 0 \), and there's no possible \(y\) that will satisfy that!
• Since \(a^0 = 1\) for any \(a\), we have \( \log_a (1) = 0\) for any base \(a\). Aka, they pass through the point \( (1,0)\).
• Similarly, since \(a^1 = a \), we have \( \log_a(a) = 1\) for any \(a\). Aka, if you take the log of its own base, you will always get 1. Also aka, they pass through the point \( (a,1) \).
• The functions \( f(x) = a^x\) and \(g(x) = \log_a x\) are inverse functions, so their graphs are mirror images across the line \(y = x\).
• While exponential functions exhibited fast growth (or decay), logarithmic functions exhibit slow growth (or decay).
• They have a vertical asymptote \(x = 0\) due to the domain issue.
• If \(a > 1\), the function is increasing (going uphill). If \( 0 < a < 1\), the function is decreasing.
• The graph \(y = \log_a x \) is the reflection of the graph \(y = \log_{1/a} x \) over the \(x\)-axis.
• A function with larger base will be growing slower (after \(x>1\) ) than a function with a smaller base. See picture below.
• Inverse Functions: \(f(x) = a^x\) and \(g(x) = \log_a x\) undo each other, i.e.
  \[ \log_a (a^x) = x, \: \text{ for all real numbers } x \quad \text{ and } \quad a^{\log_a x} = x, \: \text{ for all } x> 0 \notag \]

Log Laws

Log Law	English Translation	Caveman Translation
\( \log_a (AB) = \log_a A + \log_a B \)	The logarithm of a product of expressions (numbers, variables, whatever as long as it is being multiplied) is the sum of the logarithms of the expressions.	Multiplication inside the log can be written as addition outside the log. Used backwards, addition outside the log translates to multiplication inside the log.
\( \log_a \left( \frac{A}{B} \right) = \log_a A - \log_a B \)	The logarithm of a quotient of expressions is the difference of the logarithms of the expressions.	Division inside the log can be written as subtraction outside the log. Used backwards, subtraction outside the log translates to division inside the log. The denominator must be the input whose log term is subtracted!
\( \log_a ( A^B) = B \log_a A \)	The logarithm of an expression raised to a power is the power times the logarithm of the expression.	If the whole input is raised to a power, you can bring powers down in front as coefficients.

Note: The bases must match in order to use these laws backwards! Using them forwards is called expanding, and using them backwards is called condensing.