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Questions and Answers
What is the general form of a piecewise function and how is it applied in real-world situations?
What is the general form of a piecewise function and how is it applied in real-world situations?
The general form of a piecewise function is defined by different expressions over different intervals of the domain. It is applied in real-world situations to model scenarios that require different rules under specific conditions, such as tax brackets or pricing models.
Explain the concept of periodicity in trigonometric functions and give an example.
Explain the concept of periodicity in trigonometric functions and give an example.
Periodicity refers to the repeating nature of trigonometric functions, such as sine and cosine, which repeat every $2 heta$. For example, the sine function repeats its values every $360^ ext{o}$ or $2 ext{Ï€}$ radians.
How can you determine if a function has an inverse, and what role does the vertical line test play in this determination?
How can you determine if a function has an inverse, and what role does the vertical line test play in this determination?
A function has an inverse if each output is mapped from a unique input, which can be verified using the vertical line test. If a vertical line crosses the graph of the function more than once, the function is not one-to-one and does not have an inverse.
Describe what absolute value functions represent and their general graphical appearance.
Describe what absolute value functions represent and their general graphical appearance.
What are the key transformations of functions, and how do they affect the graph?
What are the key transformations of functions, and how do they affect the graph?
What is the general form of a polynomial function and how is its degree determined?
What is the general form of a polynomial function and how is its degree determined?
Explain how to find the real roots of a polynomial function.
Explain how to find the real roots of a polynomial function.
Define a vertical asymptote in the context of rational functions.
Define a vertical asymptote in the context of rational functions.
What distinguishes exponential growth from exponential decay?
What distinguishes exponential growth from exponential decay?
What is the general form of a logarithmic function?
What is the general form of a logarithmic function?
Describe the behavior of polynomial functions as x approaches positive or negative infinity.
Describe the behavior of polynomial functions as x approaches positive or negative infinity.
What happens in a rational function when both the numerator and denominator share a common factor?
What happens in a rational function when both the numerator and denominator share a common factor?
How do trigonometric functions relate angles to triangle sides?
How do trigonometric functions relate angles to triangle sides?
Flashcards
Periodicity of Trigonometric Functions
Periodicity of Trigonometric Functions
Trigonometric functions like sine, cosine, and tangent repeat their values after a certain interval.
Piecewise Function
Piecewise Function
A function defined by different rules for different parts of its domain.
Absolute Value Functions
Absolute Value Functions
These functions measure the distance of a number from zero, regardless of its sign.
Function Transformations
Function Transformations
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Inverse Function
Inverse Function
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Polynomial Function
Polynomial Function
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Degree of a Polynomial
Degree of a Polynomial
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Roots of a Polynomial
Roots of a Polynomial
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End Behavior of a Polynomial
End Behavior of a Polynomial
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Turning Points of a Polynomial
Turning Points of a Polynomial
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Rational Function
Rational Function
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Vertical Asymptotes
Vertical Asymptotes
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Horizontal Asymptotes
Horizontal Asymptotes
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Study Notes
Introduction to Advanced Functions
- Advanced functions encompass a broader range of functions beyond basic algebraic expressions.
- They often involve more complex mathematical operations and relationships.
- These functions can be characterized by specific properties and behaviors.
Polynomial Functions
- Polynomial functions are expressions involving variables raised to non-negative integer powers.
- General form: f(x) = anxn + an-1xn-1 +...+ a1x + a0
- Degree: The highest power of the variable (n) determines the degree of the polynomial.
- Real roots: Found by setting the function equal to zero. Roots can be found using factoring or graphical methods.
- End behavior: The behavior of the graph as x approaches positive or negative infinity is crucial. It's determined by the leading coefficient and the degree.
- Turning points: The points where the graph changes direction. Correspond to the degree and the nature of the coefficients. The number of turning points of a polynomial can be determined from the degree
Rational Functions
- Rational functions are fractions of polynomials.
- General form: f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions.
- Asymptotes: Vertical asymptotes occur where the denominator is zero. Horizontal asymptotes depend on the degrees of the numerator and denominator.
- Holes: A hole appears in the graph where both the numerator and denominator have a common factor that can be cancelled.
- Applications: Modeling real-world scenarios involving proportions or rates.
Exponential Functions
- Exponential functions involve a variable in the exponent.
- General form: f(x) = ax, where 'a' is the base.
- Growth/Decay: Exponential functions model growth or decay depending on the value of 'a'. If a > 1, it's growth; 0 < a < 1, it's decay.
- Applications: Modeling population growth, compound interest, radioactive decay, and other phenomena involving exponential change.
- Key features include: the base, the asymptote (often the x-axis), and the rate of change.
Logarithmic Functions
- Logarithmic functions are the inverse of exponential functions.
- General form: y = loga(x), where a is the base.
- Properties: Several key properties exist for logarithms, including change of base and simplifying complex logarithm expressions.
- Applications: Used to model problems involving scaling and proportions.
Trigonometric Functions
- Trigonometric functions relate angles to ratios of sides in a right triangle.
- Key functions include sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent).
- Applications: Modeling periodic phenomena, waves, angles, etc.
- Domain and range: The set of input (x) values and output (y) values that the function can accept.
- Periodicity: These functions repeat their values every certain interval.
Piecewise Functions
- Piecewise functions are defined by different expressions over different intervals of the domain.
- Defining different rules for subparts of the graph. Understanding how to graph these functions is pivotal.
- Applications to real-world situations requiring different sets of rules to reflect specific conditions.
Absolute Value Functions
- Absolute value functions describe how far a number is from zero.
- General form: f(x) = |x|.
- Graphically, they exhibit a 'V' shape. Recognising symmetry and the effect of inputs is important.
- Applications arise in problems where magnitude is of concern.
Transforming Functions
- Function transformations involve shifting, stretching, and reflecting graphs of basic functions.
- Key transformations: Vertical and horizontal shifts, stretches, and reflections across axes.
- Graphing involves understanding how transformations affect the original function.
- Recognizing these changes aids in interpreting functions in real-world contexts.
Inverse Functions
- Inverse functions "undo" the original function.
- Determining an inverse requires understanding the relationship between inputs and outputs and then switching their position.
- Involves swapping 'x' and 'y'.
- Vertical line test: Determining if a function has an inverse.
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