MATH 107 Quiz 5 Name Click here to enter text. Each question is worth 5 points. Questions 1 - 5 are multiple choice; you do not need to show your work. Short answers for Questions 6 - 10; show your work to earn full credit - highlight answers in yellow or draw a box around the answer. You can insert additional lines if needed. Leave answers in exact form unless otherwise directed to approximate the results. Write all fractions in lowest form and round decimals to hundredths. Write answers using positive exponents except when using scientific notation. Simplify all radicals and rationalize the denominators. Write complex numbers in the form a + bi. Applied problems must have the variables identified and an equation for full credit. MULTIPLE CHOICE Check the box of the one alternative that best completes the statement or answers the question. 1.) Find f(x) and g(x) such that h(x) = (f g)(x). f ( x )=12 x 2+21, g ( x ) = x 2 f ( x )= x , g ( x )=12 x +21 f ( x )= 12 x +21 , g ( x )=x 2 h ( x )= 12 x 2+ 21 f ( x )= 12 x 2 , g ( x )= 21 None of these 2.) Determine whether f 1 ( x )= f 1 ( x )= f ( x )= 6 x+5 is one-to-one. If it is one-to-one, find a formula for the inverse. x 5+ 6 x 5 x+ 6 x Not one-to-one f 1 ( x )= 5+ 6 x x None of these 4 3.) Write as an expression containing a single radical and simplify. 9 ( y+ z ) ( y+z) ( y + z ) ( y + z ) ( y + z ) 2 12 11 12 24 4 11 None of these 4.) Solve the exponential equation. 31+2 x =243 1 ( y + z ) ( y + z ) ( y + z ) 5 3 4 2 81 -2 3 6 None of these 5.) Solve the logarithmic equation. lnx=2 e2 2e ln 2 100 None of these Short answers. You must show your work to earn full credit. Highlight answers in yellow or draw a box around the answer. f ( x )= 4 x and g ( x ) =3 x 5 6.) If , find (g f)(x). 7.) Find the inverse of the function of 8.) Solve. 9.) Solve. 4x = 11 3 f ( x )= x2 . x+6+ 2x=4 2 10.) Explain why f(x) = 2x is an exponential function but f(x) = x2 is not. 3 MATH 107 Quiz 5 Name Click here to enter text. Each question is worth 5 points. Questions 1 - 5 are multiple choice; you do not need to show your work. Short answers for Questions 6 - 10; show your work to earn full credit - highlight answers in yellow or draw a box around the answer. You can insert additional lines if needed. Leave answers in exact form unless otherwise directed to approximate the results. Write all fractions in lowest form and round decimals to hundredths. Write answers using positive exponents except when using scientific notation. Simplify all radicals and rationalize the denominators. Write complex numbers in the form a + bi. Applied problems must have the variables identified and an equation for full credit. MULTIPLE CHOICE Check the box of the one alternative that best completes the statement or answers the question. 1.) Find f(x) and g(x) such that h(x) = (f g)(x). h ( x )= 12 x 2+ 21 2 f ( x )=12 x +21, g ( x ) = x f ( x )= x , g ( x )=12 x 2 +21 f ( x )= 12 x +21 , g ( x )=x 2 f ( x )= 12 x 2 , g ( x )= 21 None of these 2.) Determine whether f 1 ( x )= f 1 ( x )= f ( x )= 6 x+5 is one-to-one. If it is one-to-one, find a formula for the inverse. x 5+ 6 x 5 x+ 6 x Not one-to-one f 1 ( x )= 5+ 6 x x None of these 4 3.) Write as an expression containing a single radical and simplify. 9 ( y+ z ) ( y+z) ( y + z ) ( y + z ) ( y + z ) 2 12 11 12 24 4 11 None of these 4.) Solve the exponential equation. 3 1+2 x =243 1 ( y + z ) ( y + z ) ( y + z ) 5 3 4 2 81 -2 3 6 None of these lnx=2 5.) Solve the logarithmic equation. e2 2e ln 2 100 None of these Ans : 1 We have h ( x )=( fog )( x )h ( x )= 12 x 2 +21 Let us choose second option f ( x )= xg ( x )=12 x 2 +21 2 Then ( fog )( x )=f ( g ( x ) ) =f ( 12 x +21 ) 12 x 2+ 21 h(x) Thus Option second is correct : f ( x )= xg ( x )=12 x 2 +21 2 f ( x )= 6 , x 5 x+5 One One : Suppose f ( x 1 )=f ( x2 ) > 6 6 = , x 5x 2 5 x 1+5 x 2+ 5 1 > x 1 +5=x 2 +5= x 1=x 2 Thus f ( x ) is one one Let y=f ( x ) = 6 x +5 6 6 > x +5= = x=5+ y y > x= 5 y +6 y 2 > f 1 ( y ) = 5 y+ 6 y 5 x +6 x 1 > f ( x )= Option Second is true: f 1 ( x )= 5 x+ 6 x 3 4 ( y + z ) ( y + z ) ( y + z ) 5 3 5 4 2 4 = ( y+z) ( y+z) 2 2 4 ( y + z )3 5 2 4 + 2 3 ( y + z )4 ( y+z) ( 15+1216 ) 12 11 12 =( y + z ) 12 = ( y + z ) 11 Option second is correct : 12 ( y +z ) 11 4 1+2 x 3 =343 1 +2 x >3 5 =3 >1+2 x=5= x=2 Option fifth is correct : OF THE THESE 5 lnx=2 take exponential on bothsides we get : lnx 2 2 e =e = x=e Option first is true: 2 x=e . Short answers. You must show your work to earn full credit. Highlight answers in yellow or draw a box around the answer. 3 6.) If f ( x )= 4 x g ( x ) =3 x and 5 , find (g f)(x). Ans : 4 f ( x )= g ( x )=3 x 5 x Now ( gof ) ( x )=g ( f ( x ) ) ( 4x ) 4 3( ) x g 5 > ( gof ) ( x ) = 7.) 3072 x5 Find the inverse of the function of 3 f ( x )= x2 . Ans : 3 f ( x )= x2 3 Let f ( x )= y = x2 3 > x= y +2 Cubing on both sides we get : 3 3 ( x ) =( y +2 )3 > x=( y +2 ) 1 3 > f ( y ) =( y+ 2 ) 3 > f 1 ( x )=( x +2 )3 4 8.) x+6+ 2x=4 Solve. Ans : x+6+ 2x=4 Squaring on both sides we get : 2 ( x+6 + 2x ) =4 2 > x +6+2x+2 x+ 6 2x=16 >8+ 2 x+ 6 2x=16 >2 x+6 2x=168 > x+ 6 2x= 8 2 > x+ 6 2x=4 Again Squaring on both sides we get : 2 ( x+6 2x ) =4 2 . ( x+ 6 ) ( 2x )=16 2 >2 xx +126 x=16 2 >x 4 x=1612 2 > x +4 x + 4=0 2 > ( x +2 ) =0 > x +2=0 > x=2 9.) Solve. 4x = 11 Ans : x 4 =11 take logrithm withbase 4 on both sides x log4 4 =log 4 11 5 > x log 4 4=log4 11 > x=log4 11 x 1.730 10.) Explain why f(x) = 2x is an exponential function but f(x) = x2 is not. Ans : For f ( x )=2 x Since we know that properties of exponential function f ( x + y )=f ( x ) f ( y ) Here we seethat f ( x + y )=2 x+ y =2 x 2 y f ( x ) f ( y ) > f ( x + y ) =f ( x ) f ( y ) Hence f ( x )=2 x is exponential function For f ( x )=x 2 Here we seethat f ( x + y )=( x + y )2=x 2 + y 2+ 2 xy 2 f ( x ) f ( y )=x y 2 > f ( x + y ) f ( x ) f ( y ) Hence f ( x )=x 2 is not an exponential function . 6 MATH 107 Quiz 5 Name Click here to enter text. Each question is worth 5 points. Questions 1 - 5 are multiple choice; you do not need to show your work. Short answers for Questions 6 - 10; show your work to earn full credit - highlight answers in yellow or draw a box around the answer. You can insert additional lines if needed. Leave answers in exact form unless otherwise directed to approximate the results. Write all fractions in lowest form and round decimals to hundredths. Write answers using positive exponents except when using scientific notation. Simplify all radicals and rationalize the denominators. Write complex numbers in the form a + bi. Applied problems must have the variables identified and an equation for full credit. MULTIPLE CHOICE Check the box of the one alternative that best completes the statement or answers the question. 1.) Find f(x) and g(x) such that h(x) = (f g)(x). h ( x )= 12 x 2+ 21 2 f ( x )=12 x +21, g ( x ) = x f ( x )= x , g ( x )=12 x 2 +21 f ( x )= 12 x +21 , g ( x )=x 2 f ( x )= 12 x 2 , g ( x )= 21 None of these 2.) Determine whether f 1 ( x )= f 1 ( x )= f ( x )= 6 x+5 is one-to-one. If it is one-to-one, find a formula for the inverse. x 5+ 6 x 5 x+ 6 x Not one-to-one f 1 ( x )= 5+ 6 x x None of these 4 3.) Write as an expression containing a single radical and simplify. 9 ( y+ z ) ( y+z) ( y + z ) ( y + z ) ( y + z ) 2 12 11 12 24 4 11 None of these 4.) Solve the exponential equation. 3 1+2 x =243 1 ( y + z ) ( y + z ) ( y + z ) 5 3 4 2 81 -2 3 6 None of these lnx=2 5.) Solve the logarithmic equation. e2 2e ln 2 100 None of these Ans : 1 We have h ( x )=( fog )( x )h ( x )= 12 x 2 +21 Let us choose second option f ( x )= xg ( x )=12 x 2 +21 2 Then ( fog )( x )=f ( g ( x ) ) =f ( 12 x +21 ) 12 x 2+ 21 h(x) Thus Option second is correct : f ( x )= xg ( x )=12 x 2 +21 2 f ( x )= 6 , x 5 x+5 One One : Suppose f ( x 1 )=f ( x2 ) > 6 6 = , x 5x 2 5 x 1+5 x 2+ 5 1 > x 1 +5=x 2 +5= x 1=x 2 Thus f ( x ) is one one Let y=f ( x ) = 6 x +5 6 6 > x +5= = x=5+ y y > x= 5 y +6 y 2 > f 1 ( y ) = 5 y+ 6 y 5 x +6 x 1 > f ( x )= Option Second is true: f 1 ( x )= 5 x+ 6 x 3 4 ( y + z ) ( y + z ) ( y + z ) 5 3 5 4 2 4 = ( y+z) ( y+z) 2 2 4 ( y + z )3 5 2 4 + 2 3 ( y + z )4 ( y+z) ( 15+1216 ) 12 11 12 =( y + z ) 12 = ( y + z ) 11 Option second is correct : 12 ( y +z ) 11 4 1+2 x 3 =343 1 +2 x >3 5 =3 >1+2 x=5= x=2 Option fifth is correct : OF THE THESE 5 lnx=2 take exponential on bothsides we get : lnx 2 2 e =e = x=e Option first is true: 2 x=e . Short answers. You must show your work to earn full credit. Highlight answers in yellow or draw a box around the answer. 3 6.) If f ( x )= 4 x g ( x ) =3 x and 5 , find (g f)(x). Ans : 4 f ( x )= g ( x )=3 x 5 x Now ( gof ) ( x )=g ( f ( x ) ) ( 4x ) 4 3( ) x g 5 > ( gof ) ( x ) = 7.) 3072 x5 Find the inverse of the function of 3 f ( x )= x2 . Ans : 3 f ( x )= x2 3 Let f ( x )= y = x2 3 > x= y +2 Cubing on both sides we get : 3 3 ( x ) =( y +2 )3 > x=( y +2 ) 1 3 > f ( y ) =( y+ 2 ) 3 > f 1 ( x )=( x +2 )3 4 8.) x+6+ 2x=4 Solve. Ans : x+6+ 2x=4 Squaring on both sides we get : 2 ( x+6 + 2x ) =4 2 > x +6+2x+2 x+ 6 2x=16 >8+ 2 x+ 6 2x=16 >2 x+6 2x=168 > x+ 6 2x= 8 2 > x+ 6 2x=4 Again Squaring on both sides we get : 2 ( x+6 2x ) =4 2 . ( x+ 6 ) ( 2x )=16 2 >2 xx +126 x=16 2 >x 4 x=1612 2 > x +4 x + 4=0 2 > ( x +2 ) =0 > x +2=0 > x=2 9.) Solve. 4x = 11 Ans : x 4 =11 take logrithm withbase 4 on both sides x log4 4 =log 4 11 5 > x log 4 4=log4 11 > x=log4 11 x 1.730 10.) Explain why f(x) = 2x is an exponential function but f(x) = x2 is not. Ans : For f ( x )=2 x Since we know that properties of exponential function f ( x + y )=f ( x ) f ( y ) Here we seethat f ( x + y )=2 x+ y =2 x 2 y f ( x ) f ( y ) > f ( x + y ) =f ( x ) f ( y ) Hence f ( x )=2 x is exponential function For f ( x )=x 2 Here we seethat f ( x + y )=( x + y )2=x 2 + y 2+ 2 xy 2 f ( x ) f ( y )=x y 2 > f ( x + y ) f ( x ) f ( y ) Hence f ( x )=x 2 is not an exponential function . 6