Quiz about IB AAHL Function Review

Question 1

Define the term/unction and state a test for determining whether the graph of a relation is a function.

Accepted Answer

A function is a relation in which no two different ordered pairs have the same x-coordinate.

Accepted Answer

A graph of a relation is a function if all possible vertical lines on the graph cut the relation no more than once.

Answer

A function is a relation in which more two different ordered pairs have the same x-coordinate.

Answer

A function is a relation in which only one ordered pairs have the same x-coordinate.

Question 2

Question 2.

Accepted Answer

a).Damian={x|x>2}

Range= $$\left\{y\left|y\in R ight| ight\}$$

b).Domain=

$$\left\{x\left|\ x\ge-1,\ x e1 ight| ight\}$$

Range= $$\left\{y\left|y\in R ight| ight\}$$

Answer

a).Damian={x|x>2}

Range={y|y>0}

b).Domain=

$$\left\{x\left|\ x\ge-1\right|\right\}$$

Range= $$\left\{y\left|y\in R\right|\right\}$$

Answer

a).Damian={x|x>2}

Range= $$\left\{y\left|y\in R\right|\right\}$$

b).Domain=

$$\left\{x\left|\ x\ge-1\right|\right\}$$

Range= $$\left\{y\left|y\in R\right|\right\}$$

Answer

a).Damian={x|x>2}

Range= $$\left\{y\left|y\in R ight| ight\}$$

b).Domain=

$$\left\{x\left|\ x\ge-1,\ x e1 ight| ight\}$$

Range= $$\left\{y\left|y<0 ight| ight\}$$

Question 3

Functions f and g are defined by f : x >--+ 3x + 1 and g : x >--+ 4 - x. Find: a). f(g(x)) b). (gof)(-4)

c). $$f^{-1}\left(\frac{1}{2}\right)$$

Accepted Answer

a). -3x+13

b). 15

c).-1/6

Answer

a). -3x+13

b). 15

c).1/6

Answer

a). 3x+13

b). 15

c).1/6

Answer

a). -3x+13

b). -15

c).-1/6

Question 4

Find the equation of the resulting image when $$y=\frac{2}{x}$$ is:

a). a reflected in the y-axis

b). translated by (-1 2)

c).stretched horizontally with scale factor 3.

Accepted Answer

a). $$y=-\frac{2}{x}$$

b). $$y=\frac{2}{x+1}+2$$

c). $$y=\frac{6}{x}$$

Answer

a). $$y=\frac{2}{x}$$

b). $$y=\frac{2}{x+1}+2$$

c). $$y=\frac{6}{x}$$

Answer

a). $$y=-\frac{2}{x}$$

b). $$y=\frac{2}{x+1}+1\leftarrow$$

c). $$y=\frac{6}{x}$$

Answer

a). $$y=-\frac{2}{x}$$

b). $$y=\frac{2}{x+1}+2$$

c). $$y=-\frac{6}{x}$$

Question 5

Graph $$y=-1+2^{-x}$$ and include on your graph any axes intercepts and asymptotes.

Accepted Answer

Option B is correct

Answer

A.

Question 6

Consider the function f (x)= 2x2 +4x - 20.

a State the equation of the axis of symmetry.

b State the coordinates of the turning point.

c Write the function in the form:

i y=a(x- h)2+k ii y=a(x- p)(x- q).

Accepted Answer

a) x=-1

b).(-1,-22)

c). $$y=2^{ }\left(x+1\right)^2-22$$ $$y=2\left(x+1+\sqrt[]{11}\right)\left(x+1-\sqrt[]{11}\right)$$

Answer

a) y=-1

b).(-1,-22)

c). $$y=2^{ }\left(x+1\right)^2-22$$ $$y=2\left(x+1+\sqrt[]{11}\right)\left(x+1-\sqrt[]{11}\right)$$

Answer

a) y=-1

b).(-1,-22)

c). $$y=2^{ }\left(x+1\right)^2+22$$ $$y=2\left(x+1+\sqrt[]{11}\right)\left(x+1-\sqrt[]{11}\right)$$

Answer

a) x=-1

b).(-1,-22)

c). $$y=2^{ }\left(x+1\right)^2-22$$ $$y=2\left(x-1+\sqrt[]{11}\right)\left(x-1-\sqrt[]{11}\right)$$

Question 7

$$f\left(x\right)=e^{\left(x-1\right)}$$ write domain and range.

Accepted Answer

Domain = $$\left\{x\left|x\in R\right|\right\}$$

Range = {y I y > 0}

Answer

Domain = $$\left\{x\left|x\in R\right|\right\}$$

Range = {y I y < 0}

Answer

Domain = $$\left\{x\left|x>0\right|\right\}$$

Range = {y I y > 0}

Answer

Domain = $$\left\{x\left|x>0\right|\right\}$$

Range = {y I y < 0}

Question 8

f(x)=ln(x-2).,State the domain and range o f f , and the equation o f the

asymptote .

Accepted Answer

Domain= {x I x > 2}

Range =

$$\left\{y\left|y\in R\right|\right\}$$

x = 2 is a vertical asymptote.

Answer

Domain= {x I x < 2}

Range = {y I y >0}

x = 2 is a vertical asymptote.

Answer

Domain= {x I x < 2}

Range =

$$\left\{y\left|y\in R\right|\right\}$$

y = 2 is a horizontal asymptote.

Answer

Domain= {x I x > 2}

Range = {y I y > R}

y = 2 is a horizontal asymptote.

Question 9

$$Find\ f^{-1}\left(x\right)\ where\ f\left(x\right)=x^2+2x\ and\ x\in\left(-\infty,-1\right],find\ any\ invariant\ points.$$

Accepted Answer

(-1,-1)

Answer

(-1,1)

Answer

(1,-1)

Answer

(1,1)

Question 10

$$3x^2+x-1=0\ has\ roots\ p\ and\ q,\ find$$ $$p^2+q^2$$ .

Accepted Answer

$$\frac{7}{9}$$

Answer

$$\frac{6}{9}$$

Answer

$$\frac{8}{9}$$

Answer

$$\frac{11}{9}$$

Question 11

$$If\ f\left(x\right)=x^2+2\ and\ h\left(x\right)=3-2x,\ find\ \left(hof\right)\left(x\right).$$

Accepted Answer

$$\left(h\circ f\right)\left(x\right)=-2x^2-1$$

Answer

$$\left(h\circ f\right)\left(x\right)=2x^2-1$$

Answer

$$\left(h\circ f\right)\left(x\right)=2x^2+1$$

Answer

$$\left(h\circ f\right)\left(x\right)=-2x^2+1$$

Question 12

Find the remainder when $$2x^5-x^3+4x-1\ is\ divided\ by\ \left(x+1\right).$$

Accepted Answer

-6

Answer

-5

Answer

5

Answer

6

Question 13

$$The\ line\ with\ equation\ y=kx-2\ is\ a\ an gent\ to\ the\ quadratic\ y=3x^2+x+1.Find\ k.$$

Accepted Answer

-5 and 7

Answer

-5 and -7

Answer

5 and -7

Answer

5 and 7

Question 14

$$Find\ the\ inverse\ function\ of\ f\left(x\right)=e^{\left(2x+1\right)\ }and\ hence\ find\ f^{-1}\left(7\right).$$

Accepted Answer

$$\frac{\ln7-1}{2}$$

Answer

$$\frac{\ln7+1}{2}$$

Answer

$$\frac{-\ln7-1}{2}$$

Answer

$$\frac{-\ln7+1}{2}$$

Question 15

$$Find\ d\ if\ x+2\ is\ a\ factor\ of\ 2x^3+4x^2+dx-6.$$

Accepted Answer

-3

Answer

3

Answer

4

Answer

-4

Question 16

$$Find\ all\ quadratic\ equations\ with\ roots\ -1\pm\sqrt[]{3}i.$$

Accepted Answer

$$a(x^2+2x+4)=0\ where\ a e0.$$

Answer

$$a(x^2-x+4)=0\ where\ a e0.$$

Answer

$$a(x^2+2x-4)=0\ where\ a e0.$$

Answer

$$a(x^2+x+4)=0\ where\ a e0.$$

Question 17

$$When\ P(x)=2x^n-10x-5\ is\ divided\ by\ \left(x+2\right)\ the\ remainder\ is\ 47.\ Find\ n.$$

Accepted Answer

4

Answer

3

Answer

5

Answer

6

Question 18

$$Write\ 9x^4+4\ as\ a\ product\ of\ quadratic\ factors.$$

Accepted Answer

$$(3x^2+2i)(3x^2-2i)$$

Answer

$$(3x^2+2)(3x^2-2)$$

Answer

$$(\sqrt[]{3}x^2+2)(\sqrt[]{3}x^2-2)$$

Answer

$$(\sqrt[]{3}x^2+2i)(\sqrt[]{3}x^2-2i)$$

Question 19

$$If\ g(x)=\log2(2x-1)\ ,find\ g^{-1}(x)\ and\ hence\ evaluate\ g^{-1}(-6).$$

Accepted Answer

$$\frac{65}{128}$$

Answer

$$\frac{62}{128}$$

Answer

$$\frac{63}{128}$$

Answer

$$\frac{64}{128}$$

Question 20

$$Find\ m\ given\ that\ mx^2+(m-2)x+m=0\ has\ are\ peated\ root.$$

Accepted Answer

$$\frac{2}{3}\ and\ -2$$

Answer

$$-\frac{2}{3}\ and\ \ 2$$

Answer

$$-\frac{2}{3}\ and\ \ -2$$

Answer

$$\frac{2}{3}\ and\ \ 2$$

Question 21

$$Deter\min e\ whether\ the\ following\ functions\ are\ odd,\ even,\ or\ neither:\ \ y=x-\frac{1}{x},\ \ \ \ \ y=\cos2x.$$

Accepted Answer

odd, even

Answer

even, even

Answer

odd, odd

Answer

even, odd

Question 22

$$-2\ is\ a\ solution\ of\ x^2+bx+(b-2)=0.\ Find\ b\ and\ the\ other\ solution.$$

Accepted Answer

x=0

Answer

x=1

Answer

x=-1

Answer

x=1 and x=-1

Question 23

$$Write\ \ \frac{2x^3-11x^2+21x-8}{\left(x-2\right)^2}\ in\ Partrial\ fraction\ form.$$

Accepted Answer

$$2x-3+\frac{1}{x-2}+\frac{6}{\left(x-2\right)^2}$$

Answer

$$2x+3+\frac{1}{x-2}+\frac{6}{\left(x-2\right)^2}$$

Answer

$$-2x+3+\frac{1}{x+2}+\frac{6}{\left(x+2\right)^2}$$

Answer

$$2x-3+\frac{1}{x+2}+\frac{6}{\left(x+2\right)^2}$$

Question 24

$$Find\ a\in R\ given\ that\ 1+i\ is\ a\ zero\ of\ x^3+ax^2-4ax+6.$$

Accepted Answer

1

Answer

-1

Answer

2

Answer

-2

Question 25

$$Find\ the\ quotient\ and\ remainder\ when\ x^4+3x^2+x\ is\ divided\ by\ $$

$$:\ x+2\ \ and\ \ (x-1)^2.$$

Accepted Answer

26 and 11x-6

Answer

26 and 11x+6

Answer

-26 and 11x+6

Answer

-26 and 11x-6

Question 26

$$x^2+mx+1=0\ has\ roots\ \alpha\ and\ \beta.\ Find\ m\ if\ \alpha\beta=\frac{1}{\alpha}+\frac{1}{\beta}.$$

Accepted Answer

-1

Answer

1

Answer

2

Answer

-2

Question 27

$$Find\ the\ values\ of\ d\ such\ that\ y=dx+2\ meets\ y=x^2+3x+3$$

$$\ in\ two\ distinct\ points.$$

Accepted Answer

$$d<1\ and\ d>5$$

Answer

$$1<d<5$$

Answer

$$d<-5\ and\ d>-1$$

Answer

$$-5<d<-1$$

Question 28

$$x-1\ and\ x+3\ are\ factors\ of\ P(x)=2x^3+ax^2+bx-3.\ Find\ a\ and\ b$$

$$and\ the\ three\ zeros\ of\ P(x).$$

Accepted Answer

$$-3,\ 1\ and\ -\frac{1}{2}$$

Answer

$$3,\ 1\ and\ -\frac{1}{2}$$

Answer

$$-1,\ 3\ and\ -\frac{1}{2}$$

Answer

$$-3,\ 1\ and\ \ \frac{1}{2}$$

Question 29

$$a+2i\ is\ a\ root\ of\ z^2+bz+(a+6)=0.\ Find\ a\ and\ b\ given\ that\ a,b\ \in R.$$

Accepted Answer

$$a=-1,b=2\ or\ a=2,\ b=-4$$

Answer

$$a=1,b=2\ or\ a=-2,\ b=-4$$

Answer

$$a=-1,b=-2\ or\ a=2,\ b=4$$

Answer

$$a=1,b=2\ or\ a=2,\ b=4$$

Question 30

$$State\ the\ asymptotes\ of\ the\ graph\ with\ equation:$$

$$y-4=-\ln\left(x-2\right),\ \ \ \ \ \ y=\frac{3+x}{2x-1},\ \ \ \ \ y=2e^{\left(x-4\right)}$$

Accepted Answer

$$x=2;\ \ \ \ \ $$ $$x=\frac{1}{2,}y=\frac{1}{2};\ $$

$$\ \ \ \ y=0$$

Answer

$$x=4;\ \ \ \ \ $$ $$x=\frac{1}{2,}y=\frac{1}{2};\ $$

$$\ \ \ \ y=0$$

Answer

$$x=2;\ \ \ \ \ $$ $$x=\frac{1}{2,}y=\frac{3}{2};\ $$

$$\ \ \ \ y=0$$

Answer

$$x=4;\ \ \ \ \ $$ $$x=\frac{1}{2,}\ \ y=\frac{3}{2};\ $$

$$\ \ \ \ y=0$$

Question 31

$$Factorise\ x^3+bx^2+ax+ab\ into\ linear\ factors.$$

Accepted Answer

$$=(x+b)(x+i\sqrt[]{a})(x-i\sqrt[]{a})$$

Answer

$$=(x+b)(x^2+a)$$

Answer

$$=(x-b)\left(x^2+a\right)$$

Answer

$$=(x-b)(x+i\sqrt[]{a})(x-i\sqrt[]{a})$$

Question 32

$$A\ quartic\ polynomial\ P(x)\ has\ a\ graph\ which\ cuts\ the\ y-axis\ at\ 56,$$

$$cuts\ the\ x-axis\ at-1,\ touches\ the\ x-axis\ at\ 2\ and\ passes\ through\ $$

$$the\ point\ (1,20).\ Find\ P(x)\ in\exp anded\ form.$$

Accepted Answer

$$-4x^4+26x^3-42x^2-16x+56$$

Answer

$$-4x^4-26x^3+42x^2-16x+56$$

Answer

$$-4x^4-26x^3-42x^2+16x+56$$

Answer

$$-4x^4-26x^3-42x^2+16x-56$$

Question 33

$$Functions\ f\ and\ g\ are\ given\ by\ f\left(x\right)=e^{\left(x+1\right)}and\ g\left(x\right)=\ln x-1.$$

$$Find\ \ \left(g\circ f\right)\left(x\right)$$

Accepted Answer

x

Answer

$$e^x$$

Answer

$$\ln x$$

Answer

x-1

Question 34

$$Given\ that\ the\ graph\ of\ f(x)=mx^2+(m-l)x+2\ does\ not\ cut\ the$$

$$\ x-axis,\ find\ the\ possible\ values\ of\ m.$$

Accepted Answer

$$5-2\sqrt[]{6}

Answer

$$5-2\sqrt[]{7}

Answer

$$m<5-2\sqrt[]{7}\ or\ m>5+2\sqrt[]{7}$$

Answer

$$m<5-2\sqrt[]{6}\ or\ m>5+2\sqrt[]{6}$$

Question 35

$$When\ the\ polynomial\ P(x)\ is\ divided\ by\ (x-l)(x-2),the\ remainder$$ $$is\ 2x+3.\ What\ is\ the\ remainder\ when\ P(x)\ is\ divided\ by\ x-1?$$

Accepted Answer

5

Answer

3

Answer

4

Answer

6

Question 36

$$Solve\ for\ x:\ \left|x-1\right|\le\frac{x}{2}.$$

Accepted Answer

$$2\le x\le6$$

Answer

$$-2\le x\le6$$

Answer

$$-2\le x\le5$$

Answer

$$2\le x\le5$$

Question 37

$$Find\ m\ given\ that\ x^3+mx+m\ leaves\ a\ remainder\ of\ m\ when\ $$

$$divided\ by\ x-m.$$

Accepted Answer

$$m=0\ or\ -1$$

Answer

$$m=0\ or\ 1$$

Answer

$$m=-1\ or\ 0$$

Answer

$$m=-1\ or\ 1$$

Question 38

$$(x-1)^2\ is\ a\ factor\ of\ P(x)=x^4+ax^3+2x^2+bx-3.\ Find\ a\ and\ b.$$

Accepted Answer

$$a=-4,\ b=4$$

Answer

$$a=4,\ b=-4$$

Answer

$$a=4,\ b=4$$

Answer

$$a=-4,\ b=-4$$

Question 39

$$State\ the\ domain\ and\ range\ of\ y=f(x).$$

Accepted Answer

$$Domian:\left\{x\left|x\le3,x\in R\right|\right\}\ $$ $$Range=\left\{y\left|y\le-1\ \cup\ 0\le y<2\right|\right\}$$

Answer

$$Domian:\left\{x\left|x<3,x\in R\right|\right\}\ $$ $$Range=\left\{y\left|y\le-1\ \cup\ 0\le y<2\right|\right\}$$

Answer

$$Domian:\left\{x\left|x<3,x\in R\right|\right\}\ $$ $$Range=\left\{y\left|y<-1\ \cup\ 0

Answer

$$Domian:\left\{x\left|x\le3,x\in R\right|\right\}\ $$ $$Range=\left\{y\left|y\le-1\ \cup\ 0

Question 40

$$Suppose\ g(x)=f(\left|x\right|).$$

$$\ Is\ g(x)\ odd\ or\ even\ ?\ $$

$$Does\ g(x)\ have\ an\ inverse?$$

Accepted Answer

even;

g(x) does not have

an inverse.

Answer

even;

g(x) have

an inverse.

Answer

odd;

g(x) does not have

an inverse.

Answer

odd;

g(x) have

an inverse.

Question 41

$$x=2\ is\ a\ zero\ of\ x^3-x^2+(m+l)x+(2-m).Find\ m\ .$$

Accepted Answer

$$m=-2\ or\ 4$$

Answer

$$m=2\ or\ 4$$

Answer

$$m=-4\ or\ 2$$

Answer

$$m=-4\ or\ -2$$

Question 42

$$Suppose\ P(x)=2x^m+3x^n+P\ where\ m,n\ \in Z^+,\ p\in Z.$$

$$\ P(x)\ leaves\ a\ remainder\ of\ 6\ when\ divided\ by\ x-1,$$

$$and\ a\ remainder\ of\ 77\ when\ divided\ by\ x-2.\ Find\ m,\ n,\ and\ p.$$

Accepted Answer

$$m=5,n=2,p=1.$$

Answer

$$m=-5,\ n=2,\ p=1$$

Answer

$$m=-5,\ n=2,\ p=-1.$$

Answer

$$m=5,n=2,p=-1.$$

Question 43

$$Consider\ P(z)=6z^4+7z^3+8z^2+7z+k.Given\ that\ i\ is\ a\ zero\ of\ P(z),$$ $$find\ k\ and\ the\ remaining\ zeros\ of\ P(z).$$

Accepted Answer

$$k=2,\ the\ zeros\ of\ P(z)\ $$ $$are\pm i,-\frac{2}{3}and-\frac{1}{2}.$$

Answer

$$k=-2,\ the\ zeros\ of\ P(z)\ $$ $$are\pm i,-\frac{2}{3}and-\frac{1}{2}.$$

Answer

$$k=2,\ the\ zeros\ of\ P(z)\ $$ $$are\pm i,\ \frac{2}{3}and-\frac{1}{2}.$$

Answer

$$k=2,\ the\ zeros\ of\ P(z)\ $$ $$are\pm i,-\frac{2}{3}and\ \frac{1}{2}.$$

Question 44

$$x=a\ is\ a\ solution\ of\ 3x^3-11x^2+8x=12a.\ Find\ a.$$

Accepted Answer

$$a=0,-\frac{1}{3},or\ 4$$

Answer

$$a=0,\frac{1}{3},or\ 4$$

Answer

$$a=0,-\frac{1}{3},or\ -4$$

Answer

$$a=0,\frac{1}{3},or\ -4$$

Question 45

$$3-2i\ is\ a\ zero\ of\ P(x)=2x^3+mx^2-(m+l)x+(3-4m),\ m\in R.$$

$$Find\ m\ and\ the\ other\ two\ zeros\ of\ P(x).$$

Accepted Answer

$$m=-9,\ $$

$$zeros\ are\ 3\pm2i\ and-\frac{3}{2}$$

Answer

$$m=9,\ $$

$$zeros\ are\ -3\pm2i\ and-\frac{3}{2}$$

Answer

$$m=-9,\ $$

$$zeros\ are\ -3\pm2i\ and\ \frac{3}{2}$$

Answer

$$m=-9,\ $$

$$zeros\ are\ 3\pm2i\ and\ \frac{3}{2}$$

Question 46

$$z=a\ is\ a\ zero\ of\ P(z)=a^2z^3+z^2-a^4z-2.Find\ a\ and\ $$

$$the\ other\ zeros\ of\ P(z).$$

Accepted Answer

$$a=\pm\sqrt[]{2}$$

$$zeros\ are\pm\sqrt[]{2}and-\frac{1}{2}$$

Answer

$$a=\pm2$$

$$zeros\ are\pm\sqrt[]{2}and-\frac{1}{2}$$

Answer

$$a=\pm2$$

$$zeros\ are\pm2\ and-\frac{1}{2}$$

Answer

$$a=\pm\sqrt[]{2}$$

$$zeros\ are\pm\sqrt[]{2}and\ \frac{1}{2}$$

Question 47

$$Given\ the\ functions\ f\left(x\right)=2x-1\ and\ g\left(x\right)=2x^3,$$

$$find\ the\ function\ (f\circ g)^{-1}\left(x\right).$$

Accepted Answer

$$(fog)^{-1}\left(x\right)=\left(\frac{x+1}{4}\right)^{\frac{1}{3}}.$$

Answer

$$(fog)^{-1}\left(x\right)=\left(\frac{x-1}{4}\right)^{\frac{1}{3}}.$$

Answer

$$(fog)^{-1}\left(x\right)=\left(\frac{1-x}{4}\right)^{\frac{1}{3}}.$$

Answer

$$(fog)^{-1}\left(x\right)=-\left(\frac{x+1}{4}\right)^{\frac{1}{3}}.$$

Question 48

$$One\ zero\ of\ x^4+2x^3+8x^2+6x+15\ has\ form\ bi\ where\ b e0,b\in R.$$

$$Find\ b\ and\ all\ zeros\ of\ the\ polynomial.$$

Accepted Answer

$$b=\pm\sqrt[]{3}$$

$$zeros\ are\pm\sqrt[]{3}i,-1\pm2i$$

Answer

$$b=\pm3$$

$$zeros\ are\pm\sqrt[]{3}i,-1\pm2i$$

Answer

$$b=\pm3$$

$$zeros\ are\pm\sqrt[]{3}i,\ 1\pm2i$$

Answer

$$b=\pm\sqrt[]{3}$$

$$zeros\ are\pm\sqrt[]{3}i,\ 1\pm2i$$

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