College Algebra functions review 1 – Flashcards
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Function
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A *function* consists of three things: 1) A set called the Domain 2) A set called the Range 3) A rule which associates each element of the domain with a unique element of the range.
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Satisfy the Rule
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The coordinates of a point (a, b) are said to *satisfy the rule* of a function f if b = f(a).
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Graph of a function
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The *graph of a function* is the set of all points whose coordinates satisfy the rule of the function.
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Graph of a Function (Preferred)
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The *graph of a function* is the set of all points of the form (a, f(a)) where a is an element of the domain and f(a) is the corresponding range element.
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Zero of a Function
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A *zero of a function* f is a domain element k for which f(k) = 0.
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x-intercept
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An *x-intercept* of a graph in the Cartesian Coordinate System is a point where the graph intersects the x-axis.
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y-intercept
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A *y-intercept* of a graph in the Cartesian Coordinate System is a point where the graph intersects the y-axis.
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Zero Function
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The *zero function* z is the function defined by z(x) = 0 for all x in the domain of z.
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Constant Function
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A function f is called a *constant function* if its rule can be written as f(x) = k for some real number k.
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Linear Function
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A *linear function* is a function whose rule may be written in the form f(x) = mx + b where m and b are real numbers.
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Identity Function
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The *identity function* is the function I whose rule may be written in the form I(x) = x.
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Squaring Function
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The *squaring function* is the quadratic function f whose rule may be written in the form f(x) = x².
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Cubing Function
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The *cubing function* is the function f whose rule may be written in the form f(x) = x³.
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Quadratic Function
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A *quadratic function* is a function whose rule may be written in the form f(x) = ax² + bx + c where a, b, and c are real numbers and a is not zero.
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Reciprocal Function
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The *reciprocal function* is the function f whose rule may be written in the form shown above.
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Square Root Function
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The *square root function* is the function sqrt whose rule may be written in the form shown above.
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Absolute Value Function
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The *absolute value function* is a function abs whose rule may be written in the form abs(x) = | x |.
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Exponential Base e Function
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The *exponential base e function* is the function exp whose rule may be written in the form exp(x) = e× where e is the irrational number approximately equal to 2.718281828...
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Logarithm Base e Function
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The *logarithm base e function* is the function ln which is the inverse of the function exp.
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Piecewise Defined Function
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A *piecewise defined function* is a function whose rule is different for different intervals of its domain.
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Increasing Function
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A function f is *increasing on an interval* if, for any x?and x? in the interval, x?< x? implies f(x?) < f(x?).
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Decreasing Function
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A function f is *decreasing on an interval* if, for any x?and x? in the interval, x? f(x?).
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Even Function
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A function f is an *even function* if, f(x) = f(-x) for all domain elements x
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Odd Function
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A function f is an *odd function* if, f(x) = -f(-x) for all domain elements x.
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Sum of Functions
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The *sum of two functions* f and g with the same domain is the function named (f+g) whose rule may be written as (f+g)(x) = f(x) + g(x) for all x in the common domain.
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Difference of Functions
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The *difference of two functions* f and g with the same domain is the function named (f-g) whose rule may be written as (f-g)(x) = f(x) - g(x) for all x in the common domain.
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Product of functions
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The *product of two functions* f and g with the same domain is the function named (fg) whose rule may be written as (fg)(x) = [f(x)][g(x)] for all x in the common domain.
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One-to-One Function
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A function is called a *one-to-one function* if no element of the range is the associate of more than one domain element.
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Name for Composition of Two Functions
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The composition of a function f with a function g is a function whose name is shown above.
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Composition of Two Functions
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The composition of a function f with a function g is a function whose rule may be written in the form shown above.
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Inverse of a function
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Let f be a function with domain A and range B. Then the inverse of the function, if it exists, is a function named f?¹, with domain B and range A with the property shown above.
