Cumulative Algebra Review

    This topic is a collection of problems that might help someone understand their working knowledge of algebra.

(1) Solve


(2) Find and given and


(3) Determine whether the function is one-to-one (explain why) and if it is one-to-one then find a formula for the inverse function.


(4) Find and when and


(5) Determine whether the function is one-to-one; and if it is, then find a formula for the inverse function.


(6) (a) Convert     to an exponential equation and (b) convert to a logarithmic equation.

(7) Express in terms of sums and differences of (simpler) logarithms for


(8) Solve


(9) Solve


(10) Solve


(11) Find and given and


(12) Determine whether the function is one-to-one (explain why) and if it is one-to-one then find a formula for the inverse function.


(13) Find the domain and range of the function .


(14) Find the equation of the line that passes through and is perpendicular   to the line .


(15) Sketch the graph of the function


(16) Solve by completing the square.


(17) Complete the square to (a) find the vertex, (b) line of symmetry (c) and the maximum or minimum value, if any for .


(18) Solve .  


(19) Solve .


(20) Solve


(21) Find the inverse function of if it exists.


(22) Find the equation of the line that passes through the origin and is perpendicular to the perpendicular bisector through the points and


(23) Given find and


(24) Given and find and its domain and range.


(25) Express in standard form and use it to graph the function using transformations. Label the transformations, vertex and state whether it has a min. or max value.


(26) Find the inverse function for if it exists. Backup your assertions with a good sketch of the graph.


(27) Verify that and are inverse functions, if they are.


(28) Solve    and    if they have solutions.


(29) Explain what an inverse function is and describe its domain and range.


(30) Given state the following: domain, range, intervals of increasing and decreasing, intercepts, asymptotes. Backup your assertions with a good sketch of the graph.


(31) Find the exact values for the zeros of




(32) One of the factors of     is

    (a)          

    (b)              

    (c)        

    (d)          

    (e) none of these
    
    
(33) Simplify      

    (a)          

    (b)            

    (c)           

    (d)         

    (e) none of these    
    
    
(34) Consider      

    (a) there is no solution         

    (b) there are two solutions

    (c) there is one positive solution    

    (d) there is one negative solution
    
    
(35) The sum of the solutions to       is

    (a)       
    
    (b) no solutions      
    
    (c) less than 5        
    
    (d)  greater than 6      
    
    (e) none of the above
    
    
(36) The equation       is equivalent to

    (a)
         
    (b)       
        
    (c)       
    
    (d)       
    
    (e) none of the above
    
    
(37) The sum of the solutions to       is
    
    (a) 4      
    
    (b)         
    
    (c)         
    
    (d) no solution        
    
    (e) none of the above
    
    
(38) Solve for    

    (a)     
        
    (b)
    
    (c)         
    
    (d) none of the above
    
    
(39) The sum of the solutions of the equation       is

    (a) 19        
    
    (b) 17        
    
    (c) no solution        
    
    (d) 2        
    
    (e) none of the above
    
    
(40) Solve the inequality    .

    (a) all real numbers         
    
    (b) no solution        
    
    (c)
    
    (d)              
    
    (e) none of the above
    
    
(41) Simplify   

    (a)     
    
    (b)     
    
    (c)     
    
    (d)     
    
    (e) none of the above
    
    
(42) Simplify the expression assuming that and might be negative   

    (a)     
    
    (b) already is simplified    
    
    (c)

    (d)         
    
    (e) none of the above
    
    
(43) Given and determine and find its domain.


(44) Given find and such that



State the domain, range, line of symmetry, and absolute extrema. Sketch the graph and state the transformations used to make the graph from


(45) Given the quadratic function find the vertex, the intercepts, the intercept, and sketch the graph with the line of symmetry. Determine whether there are any absolute extrema.


(46) Given and find , and Determine the domain and range of each. Then solve the equation


(47) Sketch the graph of Draw and label the axes, state and plot the intercepts, perform tests for symmetry of the -axis, -axis and the origin. Plot enough points to show the shape of the graph.


(48) Determine all real numbers so that the distance between and is the distance between and


(49) Sketch the graph of Draw and label the axes, state and plot the intercepts, perform tests for symmetry of the -axis, -axis and the origin. Plot enough points to show the shape of the graph.


(50) Find the domain and range of the function


(51) Simplify the expression and rationalize the denominator for


(52) Express as a polynomial: .


(53) Factor the polynomial .


(54) Simplify the expression .


(55) Solve the equation .


(56) Solve the equation .


(57) Solve the inequality .


(58) Solve the inequality .


(59) Solve for given .


(60) Solve .


(61) Rationalize the denominator for


(62) Find the domain of the function


(63) Simplify


(64) For what values of is the distance between and greater than


(65) Sketch the graph of State the intercepts and provided details.


(66) Sketch the graph of and state the transformations used. Also state the function that was transformed. Label any intercepts and asymptotes.


(67) Solve the inequality where and


(68) Write the function in the form State the vertex, line of symmetry, intercepts, and extreme value. State where the function is increasing and where the function is decreasing.  


(69) Sketch the graph of and state the domain and range.

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