A Library of Functions Notes - Mrs. Vincent's Math
A Library of Functions Notes
Linear Function
• [pic] or [pic] where [pic] and y-intercept is [pic].
• The domain is the set of all real numbers.
• The range is the set of all real numbers.
• The graph has one y-intercept, [pic] unless is it is a vertical line.
• The graph has one x-intercept, unless it is a horizontal line.
• The graph is increasing if [pic], decreasing is [pic], and constant if [pic].
Identity Function
• [pic] or [pic]
• A special type of linear function.
• The domain is the set of all real numbers.
• The range is the set of all real numbers.
• Has a slope of one with a y-intercept of [pic].
• The x-coordinate equals the y-coordinate for each point.
• The graph is always increasing.
Constant Function
• [pic] or [pic]
• Y equals a number
• A special type of linear function.
• The graph is a horizontal line.
• The domain is the set of all real numbers.
• The range is the single number c (whatever number the y equals).
Squaring Function
• [pic] or [pic]
• A U-shaped curve
• The domain is the set of all real numbers.
• The range is the set of all nonnegative real numbers. [pic]
• The function is even.
• The graph has an intercept of [pic].
• The graph is decreasing for [pic] and increasing for [pic].
• The graph is symmetric with respect to the y-axis.
• The graph has a relative minimum at [pic].
Cubic Function
• [pic] or [pic]
• The domain is the set of all real numbers.
• The range is the set of all real numbers.
• The function is odd.
• The graph has an intercept at [pic].
• The graph is increasing everywhere.
• The graph is symmetric with respect to the origin.
Square Root Function
• [pic] or [pic]
• The domain is the set of all nonnegative real numbers. [pic]
• The range is the set of all nonnegative real numbers. [pic]
• The graph has an intercept at [pic].
• The graph is increasing everywhere in the domain.
Reciprocal Function
• [pic] or [pic]
• The domain is [pic].
• The range is [pic].
• The function is odd.
• The graph does not have any intercepts.
• The graph is decreasing everywhere in the domain.
• The graph is symmetric with respect to the origin.
Greatest Integer Function
• [pic] or [pic]
• “The greatest integer less than or equal to x.”
• The most famous step function.
• The domain is the set of all real numbers.
• The range is the set of all integers.
• The graph has a y-intercept at [pic] and x-intercepts in the interval from [pic].
• The graph is constant between each pair of consecutive integers.
• The graph jumps vertically one unit at each integer value.
Absolute Value Function
• [pic] or [pic]
• A V-shaped curve
• The domain is the set of all real numbers.
• The range is the set of all nonnegative real numbers. ([pic])
• The function is even.
• The graph has an intercept of [pic].
• The graph is decreasing for [pic] and increasing for [pic].
• The graph is symmetric with respect to the y-axis.
• The graph has a relative minimum at [pic].
Exponential Function
• [pic] or [pic]
• If [pic] then the function will be decreasing everywhere.
• If [pic] then the function will be increasing everywhere.
• The domain is the set of all real numbers.
• The range is all positive real numbers unless there is a translation. [pic]
• The graph has one y-intercept at [pic] unless it is translated.
• The graph has a horizontal asymptote.
• The graph does not have any x-intercepts unless there is a translation.
Logarithmic Function
Natural Logarithmic Function
• [pic] or [pic] or [pic] or [pic]
• The domain is the set of all positive real numbers. [pic]
• The range is the set of all real numbers.
• The graph is increasing everywhere it is defined.
• The graph has a vertical asymptote.
• The graph has one x-intercept at [pic].
• The graph does not have any y-intercepts.
Squared Reciprocal Function
• [pic] or [pic]
• The domain is [pic].
• The range is the set of all positive real numbers. [pic]
• The function is even.
• The graph does not have any intercepts.
• The graph is increasing for [pic] and decreasing for [pic].
• The graph is symmetric with respect to the y-axis.
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