Precalculus Study Guide: Functions, Exponents, Logarithms, and Trigonometry

Functions and Their Properties

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only non-negative integer powers of the variable.

• Definition: A polynomial function has the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$, where $n$ is a non-negative integer and coefficients $a_i$ are real numbers.

• Key Properties:

  • Exponents must be whole numbers (no fractions or negatives).

  • No variables in denominators or under roots.

• Example: $f(x) = 3x^2 + 2x + 1$ is a polynomial; $f(x) = 5/x$ is not.

Rational Functions and Asymptotes

Rational functions are quotients of polynomials. Their graphs may have vertical and horizontal asymptotes.

• Vertical Asymptote: Occurs where the denominator is zero and the numerator is nonzero. Example: For $f(x) = (x+8)/(x+11)$, the vertical asymptote is $x = -11$.

• Horizontal Asymptote: Determined by the degrees of numerator and denominator.

  • If degree numerator < degree denominator: $y = 0$

  • If degrees are equal: $y =$ (leading coefficient numerator)/(leading coefficient denominator)

  Example: For $f(x) = 5x/(2x^2 + 7)$, degree numerator < degree denominator, so $y = 0$.

Function Transformations and Reflections

Transformations include shifts, stretches, compressions, and reflections.

• Reflection over x-axis: Multiply the function by -1.

• Reflection over y-axis: Replace $x$ with $-x$.

• Example: $f(x) = -3/(x+2)$ is a reflection over the x-axis of $f(x) = 3/(x+2)$.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a imes b^x$, where $a$ is a constant and $b$ is the base.

• Parent Function: $f(x) = b^x$

• Transformations: $f(x) = b^{x-h} + k$ shifts the graph right by $h$ and up by $k$.

• Domain and Range:

  • Domain: $(- , )$

  • Range: $(0, )$ for $a > 0$

• Horizontal Asymptote: $y = k$

• Example: $g(x) = 5^x + 3$ has horizontal asymptote $y = 3$.

Evaluating Exponential Expressions

• Example: $e^4  54.60$ (rounded to nearest hundredth)

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions.

• Definition: $g(x) = ext{log}_b(x)$, where $b > 0$, $b eq 1$

• Domain: $x > 0$ for $g(x) = ext{log}_b(x)$; for $g(x) = ext{log}_b(x-3)$, domain is $x > 3$.

• Inverse Function: If $f(x) = a^x + k$, then $f^{-1}(x) = ext{log}_a(x - k)$

Solving Logarithmic Equations

• Change of Form: $ext{log}_b(x) = y ightarrow x = b^y$

• Example: $ext{log}_2(x) = -4 ightarrow x = 2^{-4} = 1/16$

Condensing Logarithmic Expressions

• Properties:

  • $ext{log}_b(x) + ext{log}_b(y) = ext{log}_b(xy)$

  • $k ext{log}_b(x) = ext{log}_b(x^k)$

  • $ext{log}_b(x) - ext{log}_b(y) = ext{log}_b(x/y)$

• Example: $(1/3)( ext{log}_3 a + ext{log}_3 b) - ext{log}_3(a + b) = ext{log}_3([ab]^{1/3}/(a + b))$

Trigonometry

Angles and Radian Measure

• Coterminal Angles: Angles that share the same terminal side. Example: $60^$ and $420^$ are coterminal because $420^ = 60^ + 360^$.

• Supplementary Angles: Two angles whose sum is $180^$. Example: Supplement of $45^$ is $135^$.

• Arc Length: $s = r heta$, where $r$ is radius and $heta$ is angle in radians. Example: For $r = 4.56$ ft, $heta = 7/8$ rad, $s = 4.56 imes 7/8$

Unit Circle and Trigonometric Values

• Coordinates: On the unit circle, $(x, y)$ corresponds to $( ext{cos} heta, ext{sin} heta)$.

• Example: $(1/2, 3/2)$, sine value is $3/2$.

• Exact Values:

  • $ext{csc} 45^ = 1/ ext{sin} 45^ = 2$

  • $ext{tan} 30^ = 1/3$

  • $ext{cot} 45^ = 1$

Special Triangles

• 30-60-90 Triangle:

  • Hypotenuse: $2x$

  • Short leg: $x$

  • Long leg: $x3$

  Example: If hypotenuse is $14$, short leg is $7$.

• Right Triangle:

  • Pythagorean Theorem: $a^2 + b^2 = c^2$

  • Example: Legs $6$and $8$, hypotenuse $c = 36 + 64 = 100 = 10$

Trigonometric Equations and Identities

• Solving Equations:

  • $ext{cos}(x) = ext{sin}(45^)$, $x = 45^$

• Period and Phase Shift:

  • Period of $y = ext{sin}(bx)$ is $2/b$

  • Example: $y = ext{sin}(3x)$, period is $2/3$

  • Phase shift: $y = ext{sin}(4x - )$, rewrite as $y = ext{sin}(4(x - /4))$

• Graph Behavior:

  • Tangent repeats every $$ units, sine every $2$ units.

  • Cosecant approaches infinity near its asymptotes.

Summary Table: Key Properties of Functions

Function Type	Domain	Range	Asymptotes
Polynomial	$(- , )$	$(- , )$	None
Rational	Excludes zeros of denominator	Varies	Vertical and/or horizontal
Exponential	$(- , )$	$(0, )$ or shifted	Horizontal
Logarithmic	$x > 0$ (or shifted)	$(- , )$	Vertical
Trigonometric	Varies	Varies	Vertical (for tan, cot, csc, sec)

Additional info:

• Some questions involve graphical analysis and transformations, which are essential for understanding function behavior in precalculus.

• Problems cover all major precalculus topics: polynomials, rational functions, exponential and logarithmic functions, trigonometry, and function transformations.