MTH-112 Quiz 1 Name: # :

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1 MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation x = define as a function of x? {(, ), (, ), (, ), (, )} (a) Is tis a function? (b) Wat is te domain of te relation?. In te grap is a function of x? (es / no) (c) Wat is te range of te relation?. Evaluate te function = x x at te following values. (a) f( ) x - -. Use te grap to find g( ). (b) f(a) (c) f( x) g( ) = x - g(x) - -. Find te domain and range of te above function g(x). Write our answers in interval notation. Domain = Range =

2 MTH- Quiz - Solutions Words in italics are for explanation purposes onl (not necessar to write in te tests or quizzes).. 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.. Evaluate te function = x x at te following values. (a) To find f( ), replace x wit. f( ) = ( ) ( ) = (8) ( 7) = 8 (b) To find f(a), replace x wit a. f(a) = (a) (a) = (a ) (8a ) = 8a a (c) To find f( x), replace x wit x. f( x) = ( x) ( x) = x + x. Does te equation x = define as a function of x? Solve te equation for in terms of x. x = = x Pick an x value; substitute, and find. If tere is onl one value, te equation is a function. Let x = : Ten = x = 9 Tere is onl one value. Yes.. In te grap is a function of x? ( 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 x - -. Use te grap to find g( ). g( ) means te coordinate of te point on te grap, for wic te x coordinate is. g( ) = x - g(x) - -. Find te domain and range of te above function g(x). Write our answers in interval notation. Domain is te set of x coordinates of te points on te grap. Domain = [, ) Range is te set of coordinates of te points on te grap. Range = [, )

3 MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Use te grap to find te following: x (a) In wic interval, if an, is te function increasing?. Find te function value of given x values. Ten grap te function. = (a) f() = (b) f() = (c) f() = { x + if x if x > (b) In wic interval, if an, is te function decreasing? (c) In wic interval, if an, is te function a constant? (d) f() = (d) Te relative minimum is (write our answer as an ordered pair): (e) Grap te function. (e) Te relative maximum is (write our answer as an ordered pair): (f) Domain: (g) Range: () Is te function graped even, odd or neiter? (even / odd / neiter) x

4 . (, ) 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.). Determine weter te function is even, odd or neiter. (a) = x + x + 7 (even / odd / neiter). It is recommended tat, for ever our ou spend in class, ou stud two ours on our own. (Tat is, if ou spend tree ours in class, ou stud six ours on our own, and so on.) Let x represent te number of ours ou spend in class, and represent te recommended number of ours of stud on our own. Write as a function of x, in te equation form. (b) = x + x + 7 (even / odd / neiter). For te function = x +, find: (a) f(x + ) (b) Te difference quotient: f(x + ) 7. Te grap of a function is given below. Grap g(x) = f(x ) on te same coordinate plane x Is te given function in problem 7 even, odd or neiter? (even / odd / neiter)

5 MTH- Quiz - Solutions Words in italics are for explanation purposes onl (not necessar to write in te tests or quizzes).. Use te grap to find te following: x (a) In wic interval, if an, is te function increasing? From left to rigt, if te grap goes up, it is increasing. Wen writing increasing, decreasing, and constant intervals, use x values (not values). In te grap, te increasing part is in blue color, and te corresponding interval on te x- axis is saded in blue color. (, ) (b) In wic interval, if an, is te function decreasing? From left to rigt, if te grap goes down, it is decreasing. In te grap, te decreasing parts are in red color, and te corresponding intervals on te x- axis are saded in red color. (, ) (, ) (c) In wic interval, if an, is te function a constant? From left to rigt, if te grap does not go up or down, it is constant. None. (d) Te relative minimum is (write our answer as an ordered pair): A relative minimum is a point on te grap were te grap canges from decreasing to increasing. (, ) (e) Te relative maximum is (write our answer as an ordered pair): A relative maximum is a point on te grap were te grap canges from increasing to decreasing. (, ) (f) Domain: Domain is te set of x coordinates of te points on te grap. Domain = (, ) (g) Range: Range is te set of coordinates of te points on te grap. Range = (, ) () Is te function graped even, odd or neiter? (even / odd / neiter ) (, ) x - (, ) (, ) Te point (, ) is on te grap. But te smmetric point about te axis, wic is (, ), is not on te grap. Terefore te

6 grap is not smmetric wit respect to te axis. 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.. Find te function value of given x values. Ten grap te function. { x + if x = if x > (a) Since te x value is less tan, use te top part of te function to find f(), tat is, = x +. f() = () + = (b) Since te x value is equal to, use te top part of te function to find f(), tat is, = x +. f() = () + = 0 (c) Since te x value is greater tan, use te bottom part of te function to find f(), tat is, =. f() = (d) Since te x value is greater tan, use te bottom part of te function to find f(), tat is, =. f() = (e) Grap te function. Separate te coordinate plane into two parts using te vertical line x = ; plot te four points found above (, ), (, 0), (, ), (, ); ten draw te graps troug tose points. Te rigt end point of te left piece must be a closed circle (because te left piece represents te grap for x ). Te left end point of te rigt piece must be on te vertical dotted line, and an open circle (because te rigt piece represents te grap for x > ) x -. (, ) 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 axis must also be on te grap. (, ). Determine weter te function is even, odd or neiter (a) = x +x+7 (even / odd / neiter ) Since tere are even exponents ( and 0), and an odd exponent (), te function is neiter even nor odd. (Te constant 7 is te same as 7x 0, and te exponent 0 is an even number.) (b) = x +x+7 (even / odd / neiter ) Since tere are odd exponents ( and ), and an even exponent (0), te function is neiter even nor odd.. For te function = x +, find: (a) To find f(x + ), replace x wit x +. f(x + ) = (x + ) + = x +

7 (b) Te difference quotient: f(x + ) x + ( x + ) = x + + x = = =. It is recommended tat, for ever our ou spend in class, ou stud two ours on our own. (Tat is, if ou spend tree ours in class, ou stud six ours on our own, and so on.) Let x represent te number of ours ou spend in class, and represent te recommended number of ours of stud on our own. Write as a function of x, in te equation form. x : te number of ours in class : te recommended number of ours of stud on our own = x 7. Te grap of a function is given below. Grap g(x) = f(x ) on te same coordinate plane. sifts te grap unit to te rigt, and sifts te grap units down x g(x) 8. Is te given function in problem 7 even, odd or neiter? ( even / odd / neiter) Te point (, ) is on te grap of. Te smmetric point about te axis, wic is (, ), is also on te grap. Terefore te grap is smmetric wit respect to te axis. Tus, te function is even.

8 MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Use te grap to find te following: x In te grap is a function of x? (es / no) x - -. For te function = x + x 9, find: (a) f(x + ) (a) In wic interval, if an, is te function increasing? (b) In wic interval, if an, is te function decreasing? (c) In wic interval, if an, is te function a constant? (d) Domain: (b) Te difference quotient: f(x + ) (e) Range: (f) Is te function graped even, odd or neiter? (even / odd / neiter) (g) Grap g(x) = f(x ) on te same coordinate plane.

9 . Grap te function. { x if x < = x + if x. Find te domain of te following functions. (Write our answers as intervals.) (a) = x x x x Domain: (b) g(x) = x 0. Determine weter te function is even, odd or neiter. (a) = x + x + (even / odd / neiter) Domain: (c) (x) = x + x + (b) = x + x + (even / odd / neiter) Domain:

10 MTH- Quiz - Solutions Words in italics are for explanation purposes onl (not necessar to write in te tests or quizzes).. Use te grap to find te following: x g(x) (a) In wic interval, if an, is te function increasing? From left to rigt, if te grap goes up, it is increasing. Wen writing increasing, decreasing, and constant intervals, use x values (not values). In te grap, te increasing part is in blue color, and te corresponding interval on te x- axis is saded in blue color. (0, ) (b) In wic interval, if an, is te function decreasing? From left to rigt, if te grap goes down, it is decreasing. In te grap, te decreasing part is in red color, and te corresponding interval on te x- axis is saded in red color. (, ) (c) In wic interval, if an, is te function a constant? From left to rigt, if te grap does not go up or down, it is constant. In te grap, te constant part is in green color, and te corresponding interval on te x- axis is saded in green color. (, 0) (d) Domain: Domain is te set of x coordinates of te points on te grap. Domain: (, ) (e) Range: Range is te set of coordinates of te points on te grap. Range: [, ) (f) Is te function graped even, odd or neiter? (even / odd / neiter ) (, ) x - (, ) (, ) Te point (, ) is on te grap. But te smmetric point about te axis, wic is (, ), is not on te grap. Terefore te grap is not smmetric wit respect to te axis. 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. (g) Grap g(x) = f(x ) on te same coordinate plane. sifts te grap units to te rigt, and sign at te beginning reflects te grap over te x- axis.

11 . In te grap is a function of x? ( 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 x - -. For te function = x + x 9, find: (a) f(x + ) To find f(x + ), replace x wit x +. f(x + ) = (x + ) + (x + ) 9 = (x + x + ) + x + 9 = x + x + + x + 9 (b) Te difference quotient: f(x + ) f(x + ) = x + x + + x + 9 (x + x 9) = x + x + + x + 9 x x + 9) = x + + (x + + ) = = x + +. Grap te function. { x if x < = x + if x Separate te coordinate plane into two parts using te vertical line x =. Find two points on eac side of te vertical line (find te values for an two x values less tan, and an two x values greater tan or equal to ). f( ) = ( ) = f(0) = (0) = f() = () + = f() = () + = Plot te four points found above (, ), (0, ), (, ), (, ); ten draw te graps troug tose points. Te rigt end point of te left piece must be on te vertical dotted line, and an open circle (because te left piece represents te grap for x < ). Te left end point of te rigt piece must be a closed circle (because te rigt piece represents te grap for x ) x Determine weter te function is even, odd or neiter. (a) = x +x + ( even / odd / neiter) Since all te exponents (, and 0) are even, te function is even.

12 (b) = x +x+ (even / odd / neiter ) Since tere are odd exponents ( and ), and an even exponent (0), te function is neiter even nor odd.. Find te domain of te following functions. (Write our answers as intervals.) (a) = x x x Te domain is all real numbers except te zeros of te bottom polnomial. Te domain is all real numbers except and. Domain: (, ) (, ) (, ) (b) g(x) = x 0 Inside te square root must be non-negative. x x Domain: (, 0] (c) (x) = x + x + x x = 0 (x )(x + ) = 0 x = x = Domain of an polnomial is all real numbers. Domain: (, )