Precalculus Study Guide: Functions, Graphs, and Polynomial/Rational Functions

Functions and Their Graphs

Introduction to Functions

Functions are fundamental objects in precalculus, describing relationships between variables. Understanding their properties and how to represent them graphically is essential for further study in mathematics.

• Definition: A function is a relation that assigns exactly one output value to each input value from a specified domain.

• Notation: Functions are often written as $f(x)$, where $x$ is the input variable.

• Domain and Range: The domain is the set of all possible input values, and the range is the set of all possible output values.

• Example: $f(x) = x^2$ has domain $(-\infty, \infty)$ and range $[0, \infty)$.

Graphs of Functions

Graphing functions helps visualize their behavior and key features.

• Intercepts: Points where the graph crosses the axes. x-intercepts occur where $f(x) = 0$; y-intercepts occur at $f(0)$.

• Symmetry: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

• Example: The graph of $f(x) = x^2$ is a parabola opening upwards, symmetric about the y-axis.

Linear Functions and Slope

Linear functions are the simplest type of functions, represented by straight lines.

• General Form: $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope: The rate of change of the function, calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Example: $f(x) = 2x + 3$ has a slope of 2 and a y-intercept of 3.

Transformations of Functions

Functions can be shifted, stretched, compressed, or reflected to create new graphs.

• Vertical Shift: $f(x) + k$ shifts the graph up by $k$ units.

• Horizontal Shift: $f(x - h)$ shifts the graph right by $h$ units.

• Reflection: $-f(x)$ reflects the graph over the x-axis.

• Example: $f(x) = (x - 2)^2 + 1$ is $f(x) = x^2$ shifted right 2 units and up 1 unit.

Composite and Inverse Functions

Combining functions and finding their inverses are important skills in precalculus.

• Composite Function: $(f \circ g)(x) = f(g(x))$.

• Inverse Function: If $f(g(x)) = x$, then $g$ is the inverse of $f$, denoted $f^{-1}(x)$.

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

Distance and Midpoint Formulas, Circles

These formulas are used to analyze geometric relationships in the coordinate plane.

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Polynomial and Rational Functions

Polynomial Functions

Polynomial functions are sums of terms with non-negative integer exponents.

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

• Degree: The highest exponent $n$ determines the degree and general shape of the graph.

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

• Example: $f(x) = x^3 - 2x^2 + x - 5$ is a cubic polynomial.

Zeros of Polynomials

Zeros (roots) are the values of $x$ where $f(x) = 0$.

• Finding Zeros: Set the polynomial equal to zero and solve for $x$.

• Multiplicity: The number of times a zero is repeated affects the graph's behavior at that point.

• Example: $f(x) = (x - 2)^2(x + 1)$ has zeros at $x = 2$ (multiplicity 2) and $x = -1$ (multiplicity 1).

Rational Functions and Their Graphs

Rational functions are ratios of polynomials and can have asymptotes and discontinuities.

• General Form: $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

• Vertical Asymptotes: Occur where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$.

• Example: $f(x) = \frac{1}{x - 3}$ has a vertical asymptote at $x = 3$.

Rational Inequalities

Solving rational inequalities involves finding where a rational expression is positive or negative.

• Steps:

  1. Set the inequality to zero.

  2. Find critical points (where numerator or denominator is zero).

  3. Test intervals between critical points.

• Example: Solve $\frac{x-2}{x+1} > 0$.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions model rapid growth or decay and have the form $f(x) = a \cdot b^x$.

• Base $b$: If $b > 1$, the function grows; if $0 < b < 1$, it decays.

• Example: $f(x) = 2^x$ is an exponential growth function.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions.

• Definition: $y = \log_b x$ means $b^y = x$.

• Properties:

  • $\log_b (xy) = \log_b x + \log_b y$

  • $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

  • $\log_b (x^r) = r \log_b x$

• Example: $\log_{10} 100 = 2$ because $10^2 = 100$.

Properties of Exponents and Logarithms

Understanding the rules for exponents and logarithms is essential for simplifying expressions and solving equations.

• Exponent Rules:

  • $a^m \cdot a^n = a^{m+n}$

  • $\frac{a^m}{a^n} = a^{m-n}$

  • $(a^m)^n = a^{mn}$

• Logarithm Rules: (see above)

• Example: Simplify $2^{x+2} = 4 \cdot 2^x$; solution: $2^{x+2} = 2^2 \cdot 2^x = 2^{x+2}$, so both sides are equal for all $x$.

Table: Comparison of Function Types

The following table summarizes key properties of linear, polynomial, rational, exponential, and logarithmic functions.

Type	General Form	Key Features	Example
Linear	$f(x) = mx + b$	Straight line, constant rate of change	$f(x) = 2x + 1$
Polynomial	$f(x) = a_nx^n + \cdots + a_0$	Curved graph, degree determines shape	$f(x) = x^3 - x$
Rational	$f(x) = \frac{P(x)}{Q(x)}$	Asymptotes, discontinuities	$f(x) = \frac{1}{x}$
Exponential	$f(x) = a \cdot b^x$	Rapid growth/decay, horizontal asymptote	$f(x) = 2^x$
Logarithmic	$f(x) = \log_b x$	Inverse of exponential, vertical asymptote	$f(x) = \log_{10} x$

Additional info: Topics such as complex numbers and quadratic functions, as well as modeling with variation, are also part of a standard precalculus curriculum but were not detailed in the provided material. Students should consult their textbook for further examples and practice problems.