Functions and Linear Equations

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Questions and Answers

What is a necessary condition for the sum of an infinite geometric series to converge to a finite value?

  • The series must consist of at least three terms.
  • The sum of the series must equal zero.
  • The common ratio must be less than 1 in absolute value. (correct)
  • The first term must be positive.

Which equation represents a circle in standard form?

  • $(x + h)^2 - (y + k)^2 = r^2$
  • $(x - h)^2 + (y - k)^2 = r^2$ (correct)
  • $(x - h)^2 - (y - k)^2 = r^2$
  • $(y - k)^2 = 4p(x - h)$

In solving systems of linear equations, what is the elimination method primarily used for?

  • To isolate a variable using substitution.
  • To eliminate one variable by adding or subtracting equations. (correct)
  • To convert equations into a matrix format.
  • To find the exact intersection point of the equations.

What is the purpose of finding the inverse of a matrix in the context of solving systems of equations?

<p>To find solutions in a more efficient manner. (A)</p>
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Which of these situations best illustrates the Fundamental Counting Principle?

<p>Choosing a shirt and pants for an outfit from different sets. (C)</p>
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What is the slope of a line perpendicular to the line described by the equation y = 3x + 2?

<p>-1/3 (B)</p>
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Which statement is true about the range of a rational function?

<p>It includes values that result from the domain excluding zero denominators. (C)</p>
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What is the degree of the polynomial P(x) = 4x^3 - 2x^2 + 5?

<p>3 (C)</p>
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What is the condition for the horizontal asymptote of a function where the degree of the numerator is less than the degree of the denominator?

<p>y = 0 (B)</p>
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For the exponential function y = 2 * 3^x, what happens to the function as x approaches negative infinity?

<p>The function approaches 0 (C)</p>
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Which of the following is a property of logarithmic functions?

<p>They can only take positive values. (C)</p>
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What defines an arithmetic sequence?

<p>Each term is derived by adding a constant to the previous term. (A)</p>
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If f(x) = x^2, what is f⁻¹(x)?

<p>√(x) (C)</p>
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Flashcards

Geometric Series

The sum of the terms in a geometric sequence.

Infinite Geometric Series

A geometric series where the sum converges to a finite value under specific conditions.

Conic Section

A curve formed by the intersection of a plane and a double cone.

Substitution Method

A method to solve a system of linear equations by finding a value for one variable and substituting it into other equations.

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Matrix

A mathematical object that represents a system of linear equations.

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Function

A relationship between inputs (domain) and outputs (range) where each input corresponds to exactly one output.

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Domain

The set of all possible input values for a function.

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Range

The set of all possible output values for a function.

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Slope

A function's growth or shrink rate. It's represented by the 'm' in the equation y = mx + b.

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Y-intercept

The point where a line crosses the y-axis. It's represented by 'b' in the equation y = mx + b.

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Polynomial Function

A finite sum of terms where each term is a variable raised to a non-negative integer power multiplied by a coefficient.

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Degree

The highest power of the variable in a polynomial.

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Roots of a polynomial

The values of 'x' that make the function equal to zero.

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Study Notes

Core Concepts

  • Functions: A relationship between inputs (domain) and outputs (range) where each input corresponds to exactly one output. Key types include linear, quadratic, polynomial, rational, exponential, logarithmic, trigonometric.
  • Domain & Range: Domain is the set of all possible input values. Range is the set of all possible output values.
  • Function Notation: f(x) represents the output of function f for input x.
  • Transformations: Shifts, stretches, and reflections of functions.
  • Inverse Functions: A function f⁻¹(x) which reverses the action of the function f(x).

Linear Functions

  • Equation: y = mx + b where m is the slope and b is the y-intercept.
  • Slope: Represents the rate of change.
  • Parallel lines have equal slopes.
  • Perpendicular lines have slopes that are negative reciprocals of each other.

Polynomial Functions

  • Defined by a finite sum of terms where each term is a variable raised to a non-negative integer power multiplied by a coefficient.
  • Degree: The highest power of the variable in the polynomial.
  • Fundamental Theorem of Algebra: A polynomial of degree n has n roots (real or complex).
  • Factors: If a is a root of P(x), then (x-a) is a factor.

Rational Functions

  • Defined as the quotient of two polynomials.
  • Domain restrictions: Avoid values that make the denominator zero.
  • Asymptotes: Lines that the graph approaches but never touches.
  • Vertical asymptotes occur at values where the denominator is zero.
  • Horizontal asymptotes are horizontal lines representing the limit as 'x' approaches infinity (positive or negative).
  • Oblique asymptotes may exist if the degree of the numerator is greater than the degree of the denominator.

Exponential and Logarithmic Functions

  • Exponential functions: y = a * bˣ. Characterized by constant growth/decay.
  • Logarithmic functions: y = logₐ(x). The inverse of exponential functions.
  • Properties of logarithms: Product, quotient, power rules. Common logarithms use base 10. Natural logarithms use base e.

Trigonometric Functions

  • Sine, cosine, tangent, cotangent, secant, cosecant.
  • Unit circle definitions: Relationships between angles and points on the unit circle.
  • Trigonometric identities: Equations that hold true for all valid input values.

Sequences and Series

  • Arithmetic sequences: Sequences with a common difference.
  • Geometric sequences: Sequences with a common ratio.
  • Arithmetic series: The sum of terms in an arithmetic sequence.
  • Geometric series: The sum of terms in a geometric sequence.
  • Infinite geometric series: Under specific conditions, the sum converges to a finite value.

Conics

  • Parabolas, ellipses, hyperbolas, circles.
  • Equations: Representing each conic section in the coordinate plane.
  • Standard forms: Identifying the center, vertex, foci, axis of symmetry.

Systems of Equations and Inequalities

  • Solving systems of linear equations: Substitution, elimination, graphing.
  • Solving systems of inequalities: Graphing the inequalities and finding the feasible region.

Matrices

  • Operations on matrices: Addition, subtraction, multiplication.
  • Matrix inverses: Existence and calculation.
  • Applications: Solving systems of equations, transformations.

Combinatorics

  • Permutations and combinations concepts.
  • Fundamental Counting Principle used to solve problems related to counting arrangements.

Further Topics

  • Limits and Continuity
  • Derivatives
  • Integrals

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