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## Intermediate Algebra Chapter 9

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**Intermediate AlgebraChapter 9**• Exponential • and • Logarithmic Functions**Intermediate Algebra 9.1-9.2**• Review of Functions**Def: Relation**• A relation is a set of ordered pairs. • Designated by: • Listing • Graphs • Tables • Algebraic equation • Picture • Sentence**Def: Function**• A function is a set of ordered pairs in which no two different ordered pairs have the same first component. • Vertical line test – used to determine whether a graph represents a function.**Defs: domain and range**• Domain: The set of first components of a relation. • Range: The set of second components of a relation**Objectives**• Determine the domain, range of relations. • Determine if relation is a function.**Intermediate Algebra 9.2**• Inverse Functions**Inverse of a function**• The inverse of a function is determined by interchanging the domain and the range of the original function. • The inverse of a function is not necessarily a function. • Designated by • and read f inverse**One-to-One function**• Def: A function is a one-to-one function if no two different ordered pairs have the same second coordinate.**Horizontal Line Test**• A function is a one-to-one function if and only if no horizontal line intersects the graph of the function at more than one point.**Objectives:**• Determine the inverse of a function whose ordered pairs are listed. • Determine if a function is one to one.**Intermediate Algebra 9.3**• Exponential Functions**Michael Crichton – The Andromeda Strain (1971)**• The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. It this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.”**Definition of Exponential Function**• If b>0 and b not equal to 1 and x is any real number, an exponential function is written as**Graphs-Determine domain, range, function, 1-1, x intercepts,**y intercepts, asymptotes**Graphs-Determine domain, range, function, 1-1, x intercepts,**y intercepts, asymptotes**Growth and Decay**• Growth: if b > 1 • Decay: if 0 < b < 1**Properties of graphs of exponential functions**• Function and 1 to 1 • y intercept is (0,1) and no x intercept(s) • Domain is all real numbers • Range is {y|y>0} • Graph approaches but does not touch x axis – x axis is asymptote • Growth or decay determined by base**Calculator Keys**• Second function of divide • Second function of LN (left side)**Property of equivalent exponents**• For b>0 and b not equal to 1**Compound Interest**• A= amount P = Principal t = time • r = rate per year • n = number of times compounded**Compound interest problem**• Find the accumulated amount in an account if $5,000 is deposited at 6% compounded quarterly for 10 years.**Objectives:**• Determine and graph exponential functions. • Use the natural base e • Use the compound interest formula.**Dwight Eisenhower – American President**• “Pessimism never won any battle.”**Intermediate Algebra 9.4,9.5,9.6**• Logarithmic Functions**Definition: Logarithmic Function**• For x > 0, b > 0 and b not equal to 1 toe logarithm of x with base b is defined by the following:**Properties of Logarithmic Function**• Domain:{x|x>0} • Range: all real numbers • x intercept: (1,0) • No y intercept • Approaches y axis as vertical asymptote • Base determines shape.**Shape of logarithmic graphs**• For b > 1, the graph rises from left to right. • For 0 < b < 1, the graphs falls from left to right.**Common Logarithmic Function The logarithmic function with**base 10**Natural logarithmic functionThe logarithmic function with a**base of e**Calculator Keys**• [LOG] • [LN]**Objective:**• Determine the common log or natural log of any number in the domain of the logarithmic function.**Change of Base Formula**• For x > 0 for any positive bases a and b**Objective**• Use the change of base formula to determine an approximation to the logarithm of a number when the base is not 10 or e.**Intermediate Algebra 10.5**• Properties • of • Logarithms**Objectives:**• Apply the product, quotient, and power properties of logarithms. • Combine and Expand logarithmic expressions**Norman Vincent Peale**• “Believe it is possible to solve your problem. Tremendous things happen to the believer. So believe the answer will come. It will.”**Intermediate Algebra 9.7**• Exponential • and • Logarithmic • Equations**Objective:**• Solve equations that have variables as exponents.