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# 1.4 Functions

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### 1.4 Functions

1. 1. CHAPTER 1 1.4 FUNCTIONS
2. 2. Into to functions  Relation- two quantities that are related to each other by some rule of correspondence  Function- a function f from a set A to B is a relation that assigns to each element x in the set A exactly one element y in the set B  Set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs)
3. 3. Characteristics of a Function from Set A to Set B  1. Each element A must be matched with and element in B  2. Some elements in B may not be matched with any element in A  3. Two or more elements in A may be matched with the same element in B  4. An element in A (the domain) cannot be matched with two different elements in B
4. 4. Four Ways to Represent a Function  1. Verbally by a sentence that describes how the input variable is related to the output variable  2. Numerically by a table or a list of ordered pairs that matches input values with output values  3. Graphically by points on a graph in a coordinate plane in which the input values are represented by the horizontal axis and the output values are represented by the vertical axis  4. Algebraically by an equation in two variables
5. 5. Function Notation Input Output Equation 2 x f ( x) f (x) 1 x •f(x) is read as the value of f at x or simply f of x •f(x) corresponds to the y-value for a given x •You can write y = f(x)
6. 6. Example of Finding a Solution  Let g(x) = - x2 + 4x + 1  g(2)  Replace x with 2 in g(x) = -x2 + 4x + 1  g(2) = -(2)2 + 4(2) + 1  = -4 + 8 + 1  g(2) = 5
7. 7. A Piecewise-Defined Function Evaluate the function when x = -1 2 x 1 x 0 f ( x) x 1 x 0 •Because x = -1 is less than 0, use f(x) = x2 + 1 to obtain •f(-1) = (-1)2 + 1 •f(-1) = 2
8. 8. The Domain of a Function  Implied Domain – set of all real numbers for which the expression is defined  Domain excludes x-values that result in division by zero f (x) 1 2 x 4  Another implied domain is one used to avoid even roots of negative numbers f ( x) x
9. 9. Summary of Function Terminology  Function- relationship between two variables such that to each value of the independent of the variable there corresponds exactly one value of the dependent variable  Function notation: y = f ( x )  f is the name of the function  y is the dependent variable  x is the independent variable  f(x) is the value of the function at x
10. 10. Terminology Cont.  Domain- is the set of all values (inputs) of the independent variable for which the function is defined  Range- set of all variables (outputs) assumed by the dependent variable  Implied Domain- consists of all real numbers for which the expression is defined
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