college algebra FINAL EXAM(solved).

Constant function f(x1) = f(x2) for any x1 and x2 in an open interval Decreasing function f(x1) > f (x2)whenever x1 < x2 on an open interval Increasing Function f(x1) < f (x2)whenever x1 < x2 on an open interval Difference of Squares a^2 -b^2 = (a+b)(a-b) Domain The set of all possible values for x Range The set of all possible values for y Function Each element in the domain corresponds to exactly one element in the range Even Function f(−x) = f(x) Odd Function f(-x)= -f(x) Piecewise Function Function that is defined by two or more equations over a specified domain Relative Maximum The point where a graph changes from increasing to decreasing Relative Minimum The point where a graph changes from decreasing to increasing Symmetric about the origin Substituting − x for x and − y for y results in the original equation Symmetric about x-axis Substituting − y for y results in the original equation Symmetric about y-axis Substituting − x for x results in the original equation Average rate of change of f f(x2) − f(x1) / x2 − x1 (from x1 to x2) Common Logarithm A logarithm whose base is 10 Composition of Functions (f o g)(x)= f(g(x)) Compound Interest Formula A=P(1+(r/n))^nt p=principle amount r=rate n=times compounded t=years Continuous Compounding Formula A=Pe^rt Exponential Function A function whose equations contain a variable in the exponent Horizontal Line Test A visual way to determine if a function has an inverse Inverse function f(f ^−1(x))= x and f^-1(f)x))= x Logarithmic Function For x > 0 and b > 0 , b ≠ 1 --- y = logbx is equivalent to b^y=x Natural exponential function f(x)=e^x Natural Logarithm Logarithm whose base is e Parallel lines Lines whose slopes are the same Perpendicular Lines Lines whose slopes are the opposite inverse of each other Point-Slope form of a line y - y1 = m(x - x1) Slope m=y2-y1/x2-x1 Change-of-Base Property logb ⁡M = log a ⁡M/log a ⁡b (for any a and b and m>0) Consistent System A system of equations with one solution Dependent system A system of equations with infinitely many solutions Inconsistent System A system of equations with no solutions Exponential equation An equation that contains a variable in an exponential expression Linear Inequality A linear equation that contains >, <, ≤, ≥ Linear system A system of linear equations Logarithmic Equation An equation that contains a variable in a logarithmic expression Nonlinear system A system of nonlinear equations One-to-one property of exponentials If b^M = b^N then M=N One-to-One Property of Logarithms If log b M = log b ⁡N then M=N Ordered triple The solution to a system of equations in three variables Power Rule log b ⁡M^p = plog b ⁡M for b, M >0, b≠1 product rule log b ⁡MN = log b ⁡M + log b ⁡N for b, M, N > 0, b≠1 quotient rule log b (M/N) = log b ⁡M − log b ⁡N for b, M >0, b≠1 Matrix An array of numbers arranged in rows and columns Order size (m x n) Square matrix If m = n, a matrix has the same number of rows as columns and is called a square matrix. Equal matrices Two matrices are equal if and only if they have the same order and corresponding elements are equal (Two matrices A and B are equal if and only if they have the same order m * n and aij = bij for i = 1, 2,..., m and j = 1, 2,..., n) Matrix Addition A + B = [aij + bij]