Functions and Their Properties

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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?

f(x) = √x Signup and view all the answers

What is the natural logarithm function?

f(x) = ln x Signup and view all the answers

What is the reciprocal function?

f(x) = 1/x Signup and view all the answers

What is the exponential function?

f(x) = e^x Signup and view all the answers

What is the sine function?

f(x) = sin x Signup and view all the answers

What is the cosine function?

f(x) = cos x Signup and view all the answers

What is the greatest integer function?

f(x) = int(x) Signup and view all the answers

What is the absolute value function?

f(x) = |x| = abs(x) Signup and view all the answers

What is the logistic function?

f(x) = 1 / (1 + e^(-x)) Signup and view all the answers

What is the domain?

All the points that 'x' can be; what you can plug into the equation Signup and view all the answers

What is the range?

All the points that 'y' can be; what you get out of the equation Signup and view all the answers

'f(x)' is the same as...?

y Signup and view all the answers

What are extrema?

Local minimum and local maximum Signup and view all the answers

When is a function even?

When it's symmetrical about the y-axis Signup and view all the answers

When is a function odd?

When it's symmetrical about the origin Signup and view all the answers

What's the formula for determining if a function is even?

f(x) = f(-x) Signup and view all the answers

What's the formula for determining if a function is odd?

f(-x) = -f(x) Signup and view all the answers

What's a clue for determining whether a function is even or odd?

Look at the exponent of the x; if it's an even number, it's an even function and vice versa. Signup and view all the answers

By looking at a graph, how can you tell if a function is even?

If you fold the graph along the y-axis and it shows symmetry, it's even. Signup and view all the answers

By looking at a graph, how can you tell if a function is odd?

If the function's graph remains unchanged when flipped upside down, it is odd. Signup and view all the answers

What are the limits for the horizontal asymptote?

f(x) = b as x → −∞ and f(x) = b as x → ∞ Signup and view all the answers

What are the limits for the vertical asymptote?

f(x) = ±∞ as x → a⁻ and f(x) = ±∞ as x → a⁺ Signup and view all the answers

When are there no horizontal asymptotes?

If the degree of the numerator is bigger than the degree of the denominator Signup and view all the answers

When will the horizontal asymptote be y=0?

If the degree of the denominator is greater than the degree of the numerator Signup and view all the answers

When is the horizontal asymptote a ratio of coefficients?

When the degrees in the numerator and denominator are the same Signup and view all the answers

When we talk about end behavior, to what are we referring?

Horizontal asymptotes Signup and view all the answers

What does end behavior show?

How a function behaves at the end Signup and view all the answers

(f + g)(x) = ________

f(x) + g(x) Signup and view all the answers

(f - g)(x) = _________

f(x) - g(x) Signup and view all the answers

(fg)(x) = __________

f(x)g(x) Signup and view all the answers

(f / g)(x) = _________

[f(x)] / [g(x)] where g(x) ≠ 0 Signup and view all the answers

What is the composition function?

(f°g)(x) = f(g(x)) Signup and view all the answers

What does a bracket mean?

It's including that number Signup and view all the answers

What do parentheses mean?

It doesn't include that number Signup and view all the answers

What are the types of discontinuity?

Jump, infinite, removable Signup and view all the answers

What should you know how to do?

Piecewise functions, scatter plots, increasing/decreasing/constant intervals, and end behavior. Signup and view all the answers

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.