Questions and Answers
What is the identity function?
f(x) = x
What is the squaring function?
f(x) = x²
What is the cubing function?
f(x) = x³
What is the square root function?
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What is the natural logarithm function?
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What is the reciprocal function?
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What is the exponential function?
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What is the sine function?
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What is the cosine function?
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What is the greatest integer function?
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What is the absolute value function?
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What is the logistic function?
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What is the domain?
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What is the range?
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'f(x)' is the same as...?
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What are extrema?
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When is a function even?
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When is a function odd?
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What's the formula for determining if a function is even?
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What's the formula for determining if a function is odd?
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What's a clue for determining whether a function is even or odd?
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By looking at a graph, how can you tell if a function is even?
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By looking at a graph, how can you tell if a function is odd?
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What are the limits for the horizontal asymptote?
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What are the limits for the vertical asymptote?
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When are there no horizontal asymptotes?
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When will the horizontal asymptote be y=0?
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When is the horizontal asymptote a ratio of coefficients?
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When we talk about end behavior, to what are we referring?
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What does end behavior show?
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(f + g)(x) = ________
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(f - g)(x) = _________
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(fg)(x) = __________
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(f / g)(x) = _________
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What is the composition function?
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What does a bracket mean?
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What do parentheses mean?
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What are the types of discontinuity?
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What should you know how to do?
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Study Notes
Functions and Definitions
- Identity function: ( f(x) = x )
- Squaring function: ( f(x) = x^2 )
- Cubing function: ( f(x) = x^3 )
- Square root function: ( f(x) = \sqrt{x} )
- Natural logarithm function: ( f(x) = \ln{x} )
- Reciprocal function: ( f(x) = \frac{1}{x} )
- Exponential function: ( f(x) = e^x )
- Sine function: ( f(x) = \sin{x} )
- Cosine function: ( f(x) = \cos{x} )
- Greatest integer function: ( f(x) = \text{int}(x) )
- Absolute value function: ( f(x) = |x| = \text{abs}(x) )
- Logistic function: ( f(x) = \frac{1}{1 + e^{-x}} )
Domain and Range
- Domain: All permissible values for 'x' in the function.
- Range: All resultant values for 'y' from the function.
Function Properties
- ( f(x) ) is equivalent to ( y ).
- Extrema include local minimums and maximums.
- A function is even if symmetric about the y-axis.
- A function is odd if symmetric about the origin.
Function Classification
- Formula for even function: ( f(x) = f(-x) )
- Formula for odd function: ( f(-x) = -f(x) )
- Clue for determining even/odd: Check the exponent of 'x'; even exponent indicates an even function, odd exponent indicates an odd function.
Graphical Symmetry
- A graph is even if it shows symmetry when folded along the y-axis.
- A graph is odd if it looks the same when flipped upside down.
Asymptotes
-
Horizontal asymptote: ( f(x) = b )
- As ( x \to -\infty ) or ( x \to \infty ).
-
Vertical asymptote:
- ( f(x) = \pm\infty ) as ( x \to a^{-} ) (from the left).
- ( f(x) = \pm\infty ) as ( x \to a^{+} ) (from the right).
Horizontal Asymptote Conditions
- No horizontal asymptotes if the numerator's degree exceeds the denominator's.
- Horizontal asymptote at ( y = 0 ) if the denominator's degree exceeds the numerator's.
- Horizontal asymptote as a ratio of coefficients when degrees of both are the same.
End Behavior
- End behavior refers to the behavior of the function as ( x ) approaches infinity or negative infinity.
- It directly relates to the existence and value of horizontal asymptotes.
Function Operations
- Sum of functions: ( (f + g)(x) = f(x) + g(x) )
- Difference of functions: ( (f - g)(x) = f(x) - g(x) )
- Product of functions: ( (fg)(x) = f(x)g(x) )
- Quotient of functions: ( (f/g)(x) = \frac{f(x)}{g(x)} ) where ( g(x) \neq 0 )
- Composition of functions: ( (f \circ g)(x) = f(g(x)) )
Interval Notation
- Bracket [ ] indicates inclusion of endpoints.
- Parentheses ( ) indicate exclusion of endpoints.
Discontinuity Types
- Jump discontinuity: Sudden changes in function values.
- Infinite discontinuity: Function approaches infinity.
- Removable discontinuity: A hole in the graph can be "fixed."
Additional Topics
- Understanding piecewise functions and scatter plots.
- Identifying increasing, decreasing, and constant intervals.
- Recognizing bounded vs. unbounded functions.
- Building new functions from existing functions.
- Determining symmetry and analyzing continuity/discontinuity along with end behavior.
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Description
Explore various mathematical functions including their definitions, domains, and ranges. This quiz covers properties of functions such as even and odd classifications, and extremas. Test your understanding of fundamental concepts in mathematics.