Name: Date: Block: FUNCTIONS TEST STUDY GUIDE

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1 Algebra STUDY GUIDE AII.6, AII.7 Functions Mrs. Grieser Name: Date: Block: Test covers: Graphing using transformations FUNCTIONS TEST STUDY GUIDE Analyzing functions, including finding domain/range in interval and/or set builder notation, identifying asymptotes, identifying intercepts, and working with composition of functions. Be able to find inverses of functions, and to determine whether the inverse of a function is a function itself. Practice Questions: ) Describe the transformations done to y = x to graph y = x+4 -. Then graph both functions. ) Write the equation of the graphs shown, using your knowledge of transformations. For each graph, identify the and transformations made. a) : b) : c) : d) : function:

2 Algebra STUDY GUIDE AII.6, AII.7 Functions Mrs. Grieser Page ) Sketch the graphs using transformations. List the and the transformations you are making. Where requested, provide asymptotes and domain/range. a) f(x) = (x+) + b) f(x) = x c) f(x) = log(x-) d) f(x) = - x+ domain/ domain/ 4) For each relation below, state the domain, range, whether the relation is a function, whether the relation is continuous or not, the zero(s) (if any), and the y-intercept(s) (if any). Supply asymptotes if requested. a) domain: b) domain:

3 Algebra STUDY GUIDE AII.6, AII.7 Functions Mrs. Grieser Page c) domain: d) domain: 5) Identify the domain and range on the real number system of the functions below in interval notation. a) f(x) = x + 5 b) f(x) = x - c) f(x) = -x + 4 d) f(x) = x 6) Find the requested composite function for the examples below. a) f(x) = x + g(x) = -x + Find (f g)(x) b) p(x) = x + h(x) = x Find h(p(x)) and p(h(x)) c) f(x) = x g(x) = x Find f(g(x)) and g(f(x)) d) f(x) = x + 5 g(x)=x h(x)=x - Find (f g h)(x) 7) Given f() =, g() =, f() = 4 and g() = 5, evaluate (f g)(). 8) Given: f(x) x, g(x) x 9, p(x) x 5, and k(x) x 5x 6 Find a) k(5) g() b) g(f()) c) g(p(x)) d) (f f)() 9) Write the inverse of the following functions. State whether the inverse is a function. Explain how you know the inverse of the function is a function. a) f(x) = x b) y = x 5 c) y = x - d) g(x) = x 7 0) Determine whether the following two functions are inverses of each other using composition of functions. Explain. a) f(x) = x 5; g(x) = 5 x b) f(x) = x ; g(x) = x + 5

4 Algebra STUDY GUIDE AII.6, AII.7 Functions Mrs. Grieser Page 4 ) What is the equation of the inverse of the function whose graph is shown? Is the inverse a Why or why not? ) Look at the graph of the functions below and determine whether the inverse of the function will be a function. Explain. a) b) c) Review questions: ) Find the product of (5 7i)( + i). 4) Find all roots (solve): x 4-6 = 5) What are the solutions to x 7 5? 6) Solve: x 4 5 x 7) Solve: 4

5 Algebra STUDY GUIDE AII.6, AII.7 Functions Mrs. Grieser Page 5 STUDY GUIDE ANSWERS ) shift left 4, down ) a) y= x ; shift down, reflect over x- axis; y = - x - b) y = x ; shift left and up; y = x c) y = log x; shift left ; y = log( x + ) d) y = x ; shift right, up, reflect over x- axis; y = -(x-) + ) a) y = x ; shift left up 4) a) D (-, ) R (-, ), function, discontinuous, zeros: 0, x0, and x, y- intercept: 0 b) D (-, )U(, ) R (-,0)U(0, ), b) y= x ; shift up function, discontinuous, zeros: none, y-intercept: hard to tell, but a negative number close to 0 x=, y=0 c) D [, ) R (-, ], not a function, continuous, zeros:, y-intercept: none d) D (-, ) R (-,7], function, continuous, zeros: -, -, 0, ; y-intercept: 0 c) y = log x; shift right ; x = ; domain (, ), range (-, ) d) y = x ; shift left, reflect over x-axis; asymptotes y = 0; domain (-, ), range (-, 0) 5) 5) a) D (-, ) R (-, ) b) D (-, ) R [-, ) c) D (-, ) R (-, 4] d) D (-, -)U(-, ) R (-, 0)U(0, ) 7) 6) a) (f g)(x) = -x +5 b) h(p(x))=x +4x+4, p(h(x))=x + c) f(g(x))=, g(f(x))= x x x d) (f g h)(x) = x+ 8) a) -6 b) 40 c) x+6 d) 9) a) y=x -, x 0; inverse is a function because original function is - 5x b) y= ; inverse is a function (same reason) 0) a) yes because f(g(x))=g(f(x))=x b) no; even though f(g(x))=g(f(x)), the compositions don t = x. ) a) yes (- function) b,c) no (not - functions) ) + i 4) x = 5, 5) x 4 6) x = - 7) x = 50 x c) y = ; inverse not a function since original function not - d) y= x ; inverse is function because g is - ) y= x 6 ; yes the inverse is a function because the original function is one-toone