MTH-112 Quiz 1 - Solutions

Transcription

1 MTH- Quiz - Solutions Words in italics are for eplanation purposes onl (not necessar to write in te tests or. Determine weter te given relation is a function. Give te domain and range of te relation. {(, ), (, ), (, ), (, )} (a) Is tis a function? Yes. Te first components do not repeat. (b) Wat is te domain of te relation? {,,, } Te first components of ordered pairs. Wen listing components use braces { }, not parenteses ( ). (c) Wat is te range of te relation? {, } Te second components of ordered pairs. Wen listing components use braces { }, not parenteses ( ).. Evaluate te function f() + at te following values. (a) To find f( ), replace wit. f( ) ( ) + ( ) (8) + ( 7) (b) To find f(a), replace wit a. f(a) (a) + (a) (a ) + (8a ) 8a + a (c) To find f( ), replace wit. f( ) ( ) + ( ). Does te equation + define as a function of? Solve te equation for in terms of. Pick an value; substitute, and solve for. If tere is onl one value, te equation defines a function. If tere are more tan one value, te equation does not define a function. Let : Yes. Tere is onl one value.. In te grap is a function of? ( es / no) If an vertical line crosses te grap at more tan one point, ten te grap is not a function. If ever vertical line crosses te grap onl once, ten te grap is a function Use te grap to find g(). - g() means te coordinate of te point on te grap, for wic te coordinate is. g() g() - -. Find te domain and range of te above function g(). Write our answers in interval notation. Domain is te set of coordinates of te point on te grap. Domain (, ] Range is te set of coordinates of te point on te grap. Range (, ]

2 MTH- Quiz - Solutions Words in italics are for eplanation purposes onl (not necessar to write in te tests or. Use te grap to find te following: For increasing, decreasing or constant intervals, alwas give te intervals (not intervals), and use parenteses ( ), not brackets [ ]. (a) In wic interval, if an, is te function increasing? From left to rigt, if te grap is going up, it is increasing. (, ) (b) In wic interval, if an, is te function decreasing? From left to rigt, if te grap is going down, it is decreasing. (, 0) (c) In wic interval, if an, is te function a constant? From left to rigt, if te grap stas at te same level, it is constant. (0, ) (d) Domain: (, ) (e) Range: [, ) (f) Is te function graped even, odd or neiter? (even / odd / neiter ) If te grap is smmetric wit respect to te ais, it is an even function. If te grap is smmetric wit respect to te origin, it is an odd function.. Determine weter te function is even, odd or neiter. (a) f() (even / odd / neiter) Since all te eponents of te variable,, and 0, are even numbers, te function is even. (Te constant 7 is te same as 7 0, and te eponent 0 is an even number.) (b) f() (even / odd / neiter) Since tere are odd eponents ( and ), and an even eponent (0), te function is neiter even nor odd.. Find te function value of given values. Ten grap te function. { if < f() + if (a) Since te value is less tan, use te top part of te function to find f(), tat is, f(). f() (b) Since te value is greater tan or equal to, use te bottom part of te function to find f(), tat is, f() +. f() () + 0 (c) Since te value is greater tan or equal to, use te bottom part of te function to find f(), tat is, f() +. f() () + (d) Grap te function

3 MTH- Quiz - Solutions Words in italics are for eplanation purposes onl (not necessar to write in te tests or. For te function f() , find: (a) To find f( + ), replace wit +. f( + ) 9( + ) f( + ) f() (b) Te difference quotient: ( ) Te grap of a function f() is given below. Grap g() f( ) on te same coordinate plane.. Find te relative minimum and relative maimum of te grap, if te eist. f() (, ) (, ) (, ) Te relative minimum is (write our answer as an ordered pair): A relative minimum is a point on te grap were te grap turns from decreasing to increasing. f() g() (, ) Te relative maimum is (write our answer as an ordered pair): A relative maimum is a point on te grap were te grap turns from increasing to decreasing. -. Is te function f() graped in problem even, odd or neiter? ( even / odd / neiter) Te point (, ) is on te grap. Te smmetric point about te ais, wic is (, ), is also on te grap. Terefore te grap is smmetric wit respect to te ais. Tus, te function is even.. (, ) is a point on te grap of an even function. Wat oter point must be on te grap? (Write our answer as an ordered pair.) Te smmetric point of (, )wit respect to ais must also be on te grap. (, ) (, ). Is te function f() graped in problem even, odd or neiter? (even / odd neiter ) Te point (, ) is on te grap. But te smmetric point about te ais, wic is (, ), is not on te grap. Terefore te grap is not smmetric wit respect to te ais. Tus, te function is not even. Te point (, ) is on te grap. But te smmetric point about te origin, wic is (, ), is not on te grap. Terefore te grap is not smmetric wit respect to te origin. Tus, te function is not odd.

4 MTH- Quiz - Solutions Words in italics are for eplanation purposes onl (not necessar to write in te tests or. For te function f() + 7, find te f( + ) f() difference quotient: f( + ) ( + ) ( + ) + 7 ( + + ) f( + ) f() ( 8 ) 8. Te grap of a function f() is given below. Grap g() f( + ) on te same coordinate plane. f() Find te domain of te following functions. (Write our answers as intervals.) (a) f() + Te domain is all real numbers ecept. (Te number makes te bottom zero.) Domain (, ) (, ) (b) g() + 0 Under te square root must be non-negative Domain [ 0, ) (c) f() + + Domain of an polnomial is all real numbers. Domain (, ). Let f() 7 + and g(). Find te following. Simplif. (a) (f + g)() is te sum of f() and g(). (f + g)() f() + g() (7 + ) + ( ) + + (b) (f g)() is te difference between f() and g(). (f g)() f() g() [7() + ] [ () () ] (c) (fg)() is te product of f() and g(). (d) (fg)() [f()] [g()] (7 + )( ) 7 8 ( ) f () is te quotient of f() and g(). g ( ) f () f() g g() 7() + () () 8

5 MTH- Quiz - Solutions (e) To find (f g)(), substitute g function in f function. (f g)() f(g()) 7( ) (f) To find (f g)(), replace wit in (f g)() 7 +. (f g)() 7() () +